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An [[Abelian group|Abelian group]] with a ring of operators. A module is a generalization of a (linear) [[Vector space|vector space]] over a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m0644701.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m0644702.png" /> is replaced by a [[Ring|ring]].
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{{TEX|done}}
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$\renewcommand{\Im}{\operatorname{Im}}$
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$\DeclareMathOperator{\Mod}{Mod}$
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$\DeclareMathOperator{\Ker}{Ker}$
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$\DeclareMathOperator{\Coker}{Coker}$
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$\DeclareMathOperator{\Coim}{Coim}$
  
Let a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m0644703.png" /> be given. An additive Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m0644704.png" /> is called a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m0644706.png" />-module if there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m0644707.png" /> whose value on a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m0644708.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m0644709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447010.png" />, written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447011.png" />, satisfies the axioms:
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An
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[[Abelian group]] with the distributive action of a ring. A module is a generalization of a (linear)
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[[vector space]] over a
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[[field]] $K$, when $K$ is replaced by a
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[[ring]].
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447012.png" />;
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Let a ring $A$ be given. An additive Abelian group $M$ is called a left $A$-module if there is a mapping $A\times M \to M$ whose value on a pair $(a, m)$, for $a \in A$, $m \in M$, written $am$, satisfies the axioms:
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447013.png" />;
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1) $a(m_1 + m_2) = am_1 + am_2$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447015.png" /> has a unit, then it is usual to require in addition that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447017.png" />. A module with this property is called unitary or unital (cf. [[Unitary module|Unitary module]]).
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2) $(a_1 + a_2)m = a_1 m + a_2 m$;
  
Right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447019.png" />-modules are defined similarly; axiom 3) is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447020.png" />. Any right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447021.png" />-module can be considered as a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447022.png" />-module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447023.png" /> anti-isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447024.png" />; hence, corresponding to any result about right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447025.png" />-modules there is a result about left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447026.png" />-modules, and conversely. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447027.png" /> is commutative, any left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447028.png" />-module can be considered as a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447029.png" />-module and the distinction between left and right modules disappears. Below only left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447030.png" />-modules are discussed.
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3) $a_1(a_2 m) = (a_1 a_2) m$. If $A$ is a
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[[ring with identity]], then it is usual to require in addition that for any $m \in M$, $1m = m$. A module with this property is called unitary or unital (cf.
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[[Unitary module|Unitary module]]).
  
The simplest examples of modules (finite Abelian groups; they are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447031.png" />-modules) were known already to C.F. Gauss as class groups of binary quadratic forms. The general notion of a module was first encountered in the 1860's till 1880's in the work of R. Dedekind and L. Kronecker, devoted to the arithmetic of algebraic number and function fields. At approximately the same time research on finite-dimensional associative algebras, in particular, group algebras of finite groups (B. Pierce, F. Frobenius), led to the study of ideals of certain non-commutative rings. At first the theory of modules was developed primarily as a theory of ideals of a ring. Only later, in the work of E. Noether and W. Krull, it was observed that it was more convenient to formulate and prove many results in terms of arbitrary modules, and not just ideals. Subsequent developments of the theory of modules were connected with the application of methods and ideas of the theory of categories (cf. [[Category|Category]]), in particular, methods of [[Homological algebra|homological algebra]].
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Right $A$-modules are defined similarly; axiom 3) is replaced by $(ma_1)a_2 = m(a_1 a_2)$. Any right $A$-module can be considered as a left $A^\text{opp}$-module over the
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[[opposite ring]] $A^\text{opp}$ anti-isomorphic to $A$; hence, corresponding to any result about right $A$-modules there is a result about left $A^\text{opp}$-modules, and conversely. When $A$ is commutative, any left $A$-module can be considered as a right $A$-module and the distinction between left and right modules disappears. Below only left $A$-modules are discussed.
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The simplest examples of modules (finite Abelian groups; they are $\ZZ$-modules) were known already to C.F. Gauss as class groups of binary quadratic forms. The general notion of a module was first encountered in the 1860's till 1880's in the work of R. Dedekind and L. Kronecker, devoted to the arithmetic of algebraic number and function fields. At approximately the same time research on finite-dimensional associative algebras, in particular, group algebras of finite groups (B. Pierce, F. Frobenius), led to the study of ideals of certain non-commutative rings. At first the theory of modules was developed primarily as a theory of ideals of a ring. Only later, in the work of E. Noether and W. Krull, it was observed that it was more convenient to formulate and prove many results in terms of arbitrary modules, and not just ideals. Subsequent developments of the theory of modules were connected with the application of methods and ideas of the theory of categories (cf.
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[[Category|Category]]), in particular, methods of
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[[Homological algebra|homological algebra]].
  
 
===Examples of modules.===
 
===Examples of modules.===
  
  
1) Any Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447032.png" /> is a module over the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447033.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447035.png" /> the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447036.png" /> is defined as the result of adding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447037.png" /> to itself <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447038.png" /> times.
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1) Any Abelian group $M$ is a module over the ring of integers $\ZZ$. For $a \in \ZZ$ and $m \in M$ the product $am$ is defined as the result of adding $m$ to itself $a$ times.
 
 
2) When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447039.png" /> is a field, the notion of a unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447040.png" />-module is exactly equivalent to the notion of a linear vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447041.png" />.
 
 
 
3) An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447042.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447043.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447044.png" /> (provided with coordinates) can be considered as a module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447045.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447046.png" />-matrices with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447047.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447049.png" /> the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447050.png" /> is defined as multiplication of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447051.png" /> by the column of coordinates of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447052.png" />.
 
  
4) An associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447053.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447054.png" />-module. Multiplication of elements of the ring by elements of the module is ordinary multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447055.png" />.
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2) When $A$ is a field, the notion of a unitary $A$-module is exactly equivalent to the notion of a linear vector space over $A$.
  
5) The set of differential forms on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447056.png" /> has the natural structure of a module over the ring of all smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447057.png" />.
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3) An $n$-dimensional vector space $V$ over a field $K$ (provided with coordinates) can be considered as a module over the ring $M_n(K)$ of all $(n\times n)$-matrices with coefficients in $K$. For $v \in V$ and $X \in M_n(K)$ the product $Xv$ is defined as multiplication of the matrix $X$ by the column of coordinates of the vector $v$.
  
6) Connected with any Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447058.png" /> is the associative ring with identity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447059.png" />, of all endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447060.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447061.png" /> has a natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447062.png" />-module structure.
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4) An associative ring (cf.
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[[Associative rings and algebras|Associative rings and algebras]]) $A$ is a left $A$-module. Multiplication of elements of the ring by elements of the module is ordinary multiplication in $A$.
  
If there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447063.png" />-module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447064.png" />, for some ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447065.png" />, then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447066.png" /> is an endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447067.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447068.png" />. Associating with the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447069.png" /> the endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447070.png" /> that it generates, one obtains a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447072.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447073.png" />. Conversely, any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447074.png" /> defines the structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447075.png" />-module on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447076.png" />. Thus, the specification of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447077.png" />-module structure on an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447078.png" /> is equivalent to the specification of a homomorphism of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447079.png" />. Such a homomorphism is also called a representation of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447080.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447081.png" /> is called a representation module. Connected with any representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447082.png" /> is a two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447083.png" />, consisting of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447084.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447085.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447086.png" />. This ideal is called the annihilator of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447087.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447088.png" />, the representation is called faithful and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447089.png" /> is called a faithful module (or faithful representation).
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5) The set of differential forms on a smooth manifold $X$ has the natural structure of a module over the ring of all smooth functions on $X$.
  
It is obvious that a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447090.png" /> can also be considered as a module over the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447091.png" />. In particular, although the definition of a module does not assume the associativity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447092.png" />, the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447093.png" /> is always associative. Therefore, in the majority of cases the discussion may be restricted to modules over associative rings. Everywhere below, unless stated otherwise, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447094.png" /> is assumed to be associative.
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6) Connected with any Abelian group $M$ is the associative ring with identity, $\End(M)$, of all endomorphisms of $M$. The group $M$ has a natural $\End(M)$-module structure.
  
==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447095.png" />-modules.==
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If there is an $A$-module structure on $M$, for some ring $A$, then the mapping $m \mapsto am$ is an endomorphism of $M$ for any $a \in A$. Associating with the element $a \in A$ the endomorphism of $M$ that it generates, one obtains a homomorphism $\phi$ of $A$ into $\End(M)$. Conversely, any homomorphism $\phi: A \to \End(M)$ defines the structure of an $A$-module on $M$. Thus, the specification of an $A$-module structure on an Abelian group $M$ is equivalent to the specification of a homomorphism of rings $\phi: A \to \End(M)$. Such a homomorphism is also called a representation of the ring $A$, and $M$ is called a representation module. Connected with any representation $\phi$ is a two-sided ideal $\Ann(M) = \Ker \phi$, consisting of the $a \in A$ such that $am = 0$ for all $m \in M$. This ideal is called the annihilator of the module $M$. When $\Ann(M) = 0$, the representation is called faithful and $M$ is called a faithful module (or faithful representation).
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447096.png" /> be a [[Group|group]]. An additive Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447097.png" /> is called a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m06447099.png" />-module if there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470100.png" /> whose value at a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470103.png" />, is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470104.png" />, and where for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470105.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470106.png" /> is an endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470107.png" />; for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470110.png" />; and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470112.png" />, where 1 is the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470113.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470114.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470115.png" /> is an automorphism of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470116.png" />.
 
  
Right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470119.png" />-modules may be defined similarly.
+
It is obvious that a module $M$ can also be considered as a module over the quotient ring $A/\Ann(M)$. In particular, although the definition of a module does not assume the associativity of $A$, the ring $A/\Ann(M)$ is always associative. Therefore, in the majority of cases the discussion may be restricted to modules over associative rings. Everywhere below, unless stated otherwise, $A$ is assumed to be associative.
  
===Examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470120.png" />-modules.===
+
==$G$-modules.==
 +
Let $G$ be a
 +
[[Group|group]]. An additive Abelian group $M$ is called a left $G$-module if there is a mapping $G\times M \to M$ whose value at a pair $(g, m)$, where $g \in G$, $m \in M$, is written as $gm$, and where for any $g \in G$ the mapping $m \mapsto gm$ is an endomorphism of $M$; for any $g_1, g_2 \in G$, $m \in M$, $(g_1 g_2)m = g_1(g_2 m)$; and for all $m \in M$, $1m = m$, where 1 is the identity of $G$. For any $g \in G$ the mapping $m \mapsto gm$ is an automorphism of the group $M$.
  
 +
Right $G$-modules may be defined similarly.
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470121.png" /> be a [[Galois extension|Galois extension]] of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470122.png" /> with [[Galois group|Galois group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470123.png" />. Then the additive and multiplicative groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470124.png" /> have the natural structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470125.png" />-modules. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470126.png" /> is an algebraic number field, then other <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470127.png" />-modules are: the additive group of the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470128.png" />, the group of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470129.png" />, the group of divisors and the [[Divisor class group|divisor class group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470130.png" />, etc. A module over a Galois group is called a Galois module.
+
===Examples of $K$-modules.===
  
2) Let an extension of an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470131.png" /> be given, that is, an [[Exact sequence|exact sequence]] of groups
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470132.png" /></td> </tr></table>
+
1) Let $k$ be a
 +
[[Galois extension|Galois extension]] of a field $G$ with
 +
[[Galois group|Galois group]] $K$. Then the additive and multiplicative groups of $K$ have the natural structure of $G$-modules. If $k$ is an algebraic number field, then other $G$-modules are: the additive group of the ring of integers of $K$, the group of units of $K$, the group of divisors and the
 +
[[Divisor class group|divisor class group]] of $K$, etc. A module over a Galois group is called a Galois module.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470133.png" /> is an Abelian [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470134.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470135.png" /> is an arbitrary group. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470136.png" /> can be given the natural structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470137.png" />-module by putting, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470139.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470140.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470141.png" /> is an inverse image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470142.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470143.png" />.
+
2) Let an extension of an Abelian group $M$ be given, that is, an
 +
[[Exact sequence|exact sequence]] of groups
  
When the group operation in the Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470144.png" /> is written multiplicatively (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470145.png" /> is the multiplicative group of a field), the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470146.png" /> is also used instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470147.png" />, that is, the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470148.png" /> is written exponentially.
+
$$1 \too M \too F \too G \too 1,$$
 +
where $M$ is an Abelian
 +
[[Normal subgroup|normal subgroup]] of $F$ and $G$ is an arbitrary group. Then $M$ can be given the natural structure of a $G$-module by putting, for $g \in G$, $m \in M$, $gm = \overline g m \overline g\inv$, where $\overline g$ is an inverse image of $g$ in $F$.
  
Let a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470149.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470150.png" /> be given. By associating with an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470151.png" /> the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470152.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470153.png" />, a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470154.png" /> into the group of invertible elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470155.png" /> is obtained. Conversely, any homomorphims of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470156.png" /> into the group of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470157.png" /> gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470158.png" /> the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470159.png" />-module.
+
When the group operation in the Abelian group $M$ is written multiplicatively (for example, if $M$ is the multiplicative group of a field), the notation $m^g$ is also used instead of $gm$, that is, the action of $G$ is written exponentially.
  
The notions of a module over a ring and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470160.png" />-module are closely connected. Namely, any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470161.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470162.png" /> can be regarded as a module over the group ring (cf. [[Group algebra|Group algebra]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470163.png" /> if the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470164.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470165.png" /> is extended linearly, that is, if one puts
+
Let a $G$-module $M$ be given. By associating with an element $g \in G$ the automorphism $m \mapsto gm$ of $M$, a homomorphism of $G$ into the group of invertible elements of the ring $\End(M)$ is obtained. Conversely, any homomorphims of $G$ into the group of invertible elements of $\End(M)$ gives $M$ the structure of a $G$-module.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470166.png" /></td> </tr></table>
+
The notions of a module over a ring and a $G$-module are closely connected. Namely, any $G$-module $M$ can be regarded as a module over the group ring (cf.
 +
[[Group algebra|Group algebra]]) $\ZZ G$ if the action of $G$ on $M$ is extended linearly, that is, if one puts
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470167.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470169.png" />. Conversely, given a unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470170.png" />-module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470171.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470172.png" /> may be regarded to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470173.png" />-module.
+
$$\left(\sum a_i g_i\right) m = \sum a_i (g_i m),$$
 +
where $a_i \in \ZZ$, $g_i \in G$, $m \in M$. Conversely, given a unitary $\ZZ G$-module structure on $M$, $M$ may be regarded to be a $G$-module.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470174.png" /> is simultaneously a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470175.png" />-module over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470176.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470177.png" />-module, where the action of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470178.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470179.png" /> commutes with the action of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470180.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470181.png" /> may be given the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470182.png" />-module by linearly extending the action from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470183.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470184.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470185.png" /> is a linear vector space over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470186.png" />, then the specification of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470187.png" />-module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470188.png" /> is equivalent to giving a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470189.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470190.png" />.
+
When $M$ is simultaneously a $K$-module over a commutative ring $K$ and a $G$-module, where the action of the elements of $G$ on $M$ commutes with the action of the elements of $K$, then $M$ may be given the structure of a $KG$-module by linearly extending the action from $G$ to $KG$. For example, if $V$ is a linear vector space over a field $K$, then the specification of a $KG$-module structure on $V$ is equivalent to giving a representation of $G$ in $V$.
  
Using the standard involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470191.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470192.png" />, any left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470193.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470194.png" /> can be made into a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470195.png" />-module by putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470196.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470197.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470198.png" />. Similarly, any right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470199.png" />-module can be made into a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470200.png" />-module.
+
Using the standard involution $g \mapsto g\inv$ in $G$, any left $G$-module $M$ can be made into a right $G$-module by putting $mg = g\inv m$ for $m \in M$, $g \in G$. Similarly, any right $G$-module can be made into a left $G$-module.
  
 
==Modules over a Lie algebra.==
 
==Modules over a Lie algebra.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470201.png" /> be a [[Lie algebra|Lie algebra]] over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470202.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470203.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470204.png" />-module. The specification of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470205.png" />-module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470206.png" /> consists of the specification of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470207.png" />-endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470208.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470209.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470210.png" />, where the axiom
+
Let $\lieg$ be a
 
+
[[Lie algebra|Lie algebra]] over a commutative ring $K$ and let $M$ be a $K$-module. The specification of a $\lieg$-module structure on $M$ consists of the specification of a $K$-endomorphism $m \mapsto gm$ of the group $M$ for each $g \in \lieg$, where the axiom
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470211.png" /></td> </tr></table>
 
  
holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470212.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470213.png" />. This definition differs from that of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470214.png" />-module given earlier. Giving a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470215.png" />-module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470216.png" /> is equivalent to giving a Lie algebra homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470217.png" /> into the Lie algebra of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470218.png" />. A module over a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470219.png" /> may also be regarded as a module in the usual sense over the [[Universal enveloping algebra|universal enveloping algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470220.png" />.
+
$$[g_1, g_2] m = g_1(g_2 m) - g_2(g_1 m)$$
 +
holds for $g_1, g_2 \in \lieg$, $m \in M$. This definition differs from that of an $A$-module given earlier. Giving a $\lieg$-module structure on $M$ is equivalent to giving a Lie algebra homomorphism of $\lieg$ into the Lie algebra of the ring $\End(M)$. A module over a Lie algebra $\lieg$ may also be regarded as a module in the usual sense over the
 +
[[Universal enveloping algebra|universal enveloping algebra]] of $\lieg$.
  
 
==Constructions in the theory of modules.==
 
==Constructions in the theory of modules.==
Starting from a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470221.png" />-module it is possible to obtain new <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470222.png" />-modules by a number of standard constructions. Thus, with any module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470223.png" /> is associated the [[Lattice|lattice]] of its submodules. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470224.png" /> is considered as left module over itself, then its left submodules are precisely the left ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470225.png" />. A number of important types of modules are defined in terms of the lattice of submodules. For example, the condition for termination of a descending (ascending) chain of submodules defines Artinian modules (respectively, Noetherian modules, cf. [[Artinian module|Artinian module]]; [[Noetherian module|Noetherian module]]). The condition for absence of non-trivial submodules, that is, submodules other than 0 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470226.png" />, defines irreducible or simple modules (cf. [[Irreducible module|Irreducible module]]).
+
Starting from a given $A$-module it is possible to obtain new $A$-modules by a number of standard constructions. Thus, with any module $M$ is associated the
 +
[[Lattice|lattice]] of its submodules. For example, if $A$ is considered as left module over itself, then its left submodules are precisely the left ideals in $A$. A number of important types of modules are defined in terms of the lattice of submodules. For example, the condition for termination of a descending (ascending) chain of submodules defines Artinian modules (respectively, Noetherian modules, cf.
 +
[[Artinian module|Artinian module]];
 +
[[Noetherian module|Noetherian module]]). The condition for absence of non-trivial submodules, that is, submodules other than 0 or $M$, defines irreducible or simple modules (cf.
 +
[[Irreducible module|Irreducible module]]).
  
For a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470227.png" /> and any submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470228.png" />, the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470229.png" /> can be given the structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470230.png" />-module. This module is called the quotient module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470231.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470232.png" />.
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For a module $M$ and any submodule $N$, the quotient group $M/N$ can be given the structure of an $A$-module. This module is called the quotient module of $M$ over $N$.
  
A homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470233.png" />-modules is defined as an Abelian group [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470234.png" /> commuting with multiplication by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470235.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470236.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470237.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470238.png" />. If two homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470239.png" /> are given, then their sum, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470240.png" />, is again a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470241.png" />-modules. This addition gives an Abelian group structure to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470242.png" /> of all homomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470243.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470244.png" />. For any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470245.png" /> the submodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470246.png" /> (the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470247.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470248.png" /> (the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470249.png" />), and also the quotient modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470250.png" /> (the cokernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470251.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470252.png" /> (the coimage of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470253.png" />) are defined. The modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470254.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470255.png" /> are canonically isomorphic and therefore usually identified. For example, for any left ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470256.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470257.png" /> the quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470258.png" /> is defined. The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470259.png" /> is irreducible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470260.png" /> is a maximal left ideal (cf. [[Maximal ideal|Maximal ideal]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470261.png" /> is an irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470262.png" />-module not annihilating the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470263.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470264.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470265.png" /> for some maximal left ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470266.png" />.
+
A homomorphism of $A$-modules is defined as an Abelian group
 +
[[Homomorphism|homomorphism]] $f : M \to N$ commuting with multiplication by elements of $A$, that is, $f(am) = af(m)$ for all $m \in M$, $a \in A$. If two homomorphisms $f_1, f_2 : M \to N$ are given, then their sum, defined by $(f_1 + f_2)(m) = f_1(m) = f_2(m)$, is again a homomorphism of $A$-modules. This addition gives an Abelian group structure to the set $\Hom_A(M, N)$ of all homomorphisms of $M$ into $N$. For any homomorphism $f : M \to N$ the submodules $\Ker f$ (the kernel of $f$) and $\Im f$ (the image of $f$), and also the quotient modules $\Coker f = N / \Im f$ (the cokernel of $f$) and $\Coim f = M/\Ker f$ (the coimage of $f$) are defined. The modules $\Im f$ and $\Coim f$ are canonically isomorphic and therefore usually identified. For example, for any left ideal $J$ of $A$ the quotient module $A/J$ is defined. The module $A/J$ is irreducible if and only if $J$ is a maximal left ideal (cf.
 +
[[Maximal ideal|Maximal ideal]]). If $M$ is an irreducible $A$-module not annihilating the ring $A$, then $M$ is isomorphic to $A/J$ for some maximal left ideal $J$.
  
For any family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470267.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470268.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470269.png" /> runs through some index set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470270.png" />, the direct sum and direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470271.png" /> exist in the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470272.png" />-modules. Here an element of the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470273.png" /> may be interpreted as a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470274.png" /> the components of which are indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470275.png" /> and where for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470276.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470277.png" />. The sum of such vectors and their multiplication by elements of the ring are defined componentwise. The direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470278.png" /> of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470279.png" /> can be interpreted as the submodule of the direct product consisting of the vectors all components of which, except for finitely many, are equal to zero.
+
For any family of $A$-modules $\{M_i\}$, where $i$ runs through some index set $J$, the direct sum and direct product of $\{M_i\}$ exist in the category of $A$-modules. Here an element of the direct product $\prod_{i \in J} M_i$ may be interpreted as a vector $(ldots, m_i, \ldots)$ the components of which are indexed by $J$ and where for each $i$, $m_i \in M_i$. The sum of such vectors and their multiplication by elements of the ring are defined componentwise. The direct sum $\sum_{i \in J} M_i$ of the family $\{M_i\}$ can be interpreted as the submodule of the direct product consisting of the vectors all components of which, except for finitely many, are equal to zero.
  
For a projective (inductive) system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470280.png" />-modules the projective (inductive) limit of this system can be naturally equipped with the structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470281.png" />-module. The direct product and direct sum may be considered as special cases of the notions of a projective and an inductive limit.
+
For a projective (inductive) system of $A$-modules the projective (inductive) limit of this system can be naturally equipped with the structure of an $A$-module. The direct product and direct sum may be considered as special cases of the notions of a projective and an inductive limit.
  
 
==Generators and relations.==
 
==Generators and relations.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470282.png" /> be a subset of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470283.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470284.png" />. The submodule generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470285.png" /> is the intersection of the submodules of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470286.png" /> which contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470287.png" />. If this submodule coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470288.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470289.png" /> is called a family (system) of generators of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470290.png" />. A module admitting a finite family of generators is called a finitely-generated module. For example, in a Noetherian ring any ideal is a finitely-generated module. A direct sum of a finite number of finitely-generated modules is again finitely generated. Any quotient module of a finitely-generated module is also finitely generated. For the construction of a system of generators for a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470291.png" />, Nakayama's lemma often turns out to be useful: For any ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470292.png" /> contained in the radical of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470293.png" /> the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470294.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470295.png" />. In particular, under the conditions of Nakayama's lemma elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470296.png" /> form a system of generators for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470297.png" /> if their images generate the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470298.png" />. This is used particularly often when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470299.png" /> is a [[Local ring|local ring]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470300.png" /> is the maximal ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470301.png" />.
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Let $X$ be a subset of an $A$-module $M$. The submodule generated by $X$ is the intersection of the submodules of $M$ which contain $X$. If this submodule coincides with $M$, then $X$ is called a family (system) of generators of the module $M$. A module admitting a finite family of generators is called a finitely-generated module. For example, in a Noetherian ring any ideal is a finitely-generated module. A direct sum of a finite number of finitely-generated modules is again finitely generated. Any quotient module of a finitely-generated module is also finitely generated. For the construction of a system of generators for a module $M$, the
 +
[[Nakayama lemma]] often turns out to be useful: For any ideal $\frakA$ contained in the radical of a ring $A$ the condition $\frakA M = M$ implies $M = 0$. In particular, under the conditions of Nakayama's lemma elements $m_1, \ldots, m_r$ form a system of generators for $M$ if their images generate the module $M/\frakA M$. This is used particularly often when $A$ is a
 +
[[Local ring|local ring]] and $\frakA$ is the maximal ideal in $A$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470302.png" /> be a module with system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470303.png" />. Then a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470304.png" /> defines an epimorphism of the free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470305.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470306.png" /> with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470307.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470308.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470309.png" /> can be defined as the set of formal finite sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470310.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470311.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470312.png" /> is extended from the generators to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470313.png" /> by linearity). The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470314.png" /> are called relations between the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470315.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470316.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470317.png" /> can be represented as a quotient module of a finitely-generated free module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470318.png" /> so that the module of relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470319.png" /> is also finitely generated, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470320.png" /> is called a finitely-presented module. For example, over a Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470321.png" /> any finitely-generated module is finitely presented. In general, being finitely generated does not imply being finitely presented.
+
Let $M$ be a module with system of generators $\{x_i\}_{i \in J}$. Then a mapping $\phi : y_i \to x_i$ defines an epimorphism of the free $A$-module $F$ with generators $\{y_i\}_{i \in I}$ onto $M$ ($F$ can be defined as the set of formal finite sums $\sum a_i y_i$, $a_i \in A$, and $\phi$ is extended from the generators to $F$ by linearity). The elements of $R = \Ker \phi$ are called relations between the generators $\{x_i\}$ of $M$. If $M$ can be represented as a quotient module of a finitely-generated free module $F$ so that the module of relations $R$ is also finitely generated, then $M$ is called a finitely-presented module. For example, over a Noetherian ring $A$ any finitely-generated module is finitely presented. In general, being finitely generated does not imply being finitely presented.
  
 
==Change of rings.==
 
==Change of rings.==
There are standard constructions which allow an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470322.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470323.png" /> to be considered as a module over some other ring. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470324.png" /> be a homomorphism of rings. Then, putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470325.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470326.png" /> can be considered as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470327.png" />-module. The resulting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470328.png" />-module is said to be obtained by base change or, in particular in the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470329.png" /> is a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470330.png" />, by restriction of scalars. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470331.png" /> is a unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470332.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470333.png" /> takes the identity to the identity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470334.png" /> becomes a unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470335.png" />-module.
+
There are standard constructions which allow an $A$-module $M$ to be considered as a module over some other ring. For example, let $\phi : B \to A$ be a homomorphism of rings. Then, putting $bm = \phi(b) m$, $M$ can be considered as a $B$-module. The resulting $B$-module is said to be obtained by base change or, in particular in the case that $B$ is a subring of $A$, by restriction of scalars. If $M$ is a unitary $A$-module and $\phi$ takes the identity to the identity, $M$ becomes a unitary $B$-module.
  
Let a ring homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470336.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470337.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470338.png" /> be given. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470339.png" /> may be given the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470340.png" />-module (cf. [[Bimodule|Bimodule]]) by putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470341.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470342.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470343.png" />, and the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470344.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470345.png" /> can be considered. One says that this module is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470346.png" /> by extension of scalars.
+
Let a ring homomorphism $\phi: A \to B$ and an $A$-module $M$ be given. Then $B$ may be given the structure of a $(B, A)$-module (cf.
 +
[[Bimodule|Bimodule]]) by putting $ba = b\phi(a)$ for $b\in B$, $a \in A$, and the left $B$-module $B \tensor_A M$ can be considered. One says that this module is obtained from $M$ by extension of scalars.
  
 
==The category of modules.==
 
==The category of modules.==
The class of all modules over a given ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470347.png" /> with homomorphisms of modules as morphisms forms an [[Abelian category|Abelian category]], denoted, for instance, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470348.png" />-mod or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470349.png" />. The most important functors defined on this category are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470350.png" /> (homomorphism) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470351.png" /> (tensor product). The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470352.png" /> takes values in the category of Abelian groups and associates to a pair of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470353.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470354.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470355.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470356.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470357.png" /> the mappings
+
The class of all modules over a given ring $A$ with homomorphisms of modules as morphisms forms an
 
+
[[Abelian category|Abelian category]], denoted, for instance, by $A$-mod or $\Mod_A$. The most important functors defined on this category are $\Hom$ (homomorphism) and $\tensor$ (tensor product). The functor $\Hom$ takes values in the category of Abelian groups and associates to a pair of $A$-modules $M, N$ the group $\Hom_A(M, N)$. For $f : M_1 \to M$ and $\phi : N \to N_1$ the mappings
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470358.png" /></td> </tr></table>
 
  
 +
$$f' : \Hom_A(M, N) \to \Hom_A(M_1, N)$$
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470359.png" /></td> </tr></table>
+
$$\phi' : \Hom_A(M, N) \to \Hom_A(M, N_1)$$
 
+
are defined in the obvious way; that is, the functor $\Hom$ is contravariant in its first argument and covariant in the second. When $M$ or $N$ carry a bimodule structure, the group $\Hom_A(M,N)$ has an additional module structure. If $N$ is an $(A, B)$-module, $\Hom_A(M,N)$ is a right $B$-module and if $M$ is an $(A,B)$-module, then $\Hom_A(M, N)$ is a left $B$-module.
are defined in the obvious way; that is, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470360.png" /> is contravariant in its first argument and covariant in the second. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470361.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470362.png" /> carry a bimodule structure, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470363.png" /> has an additional module structure. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470364.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470365.png" />-module, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470366.png" /> is a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470367.png" />-module and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470368.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470369.png" />-module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470370.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470371.png" />-module.
 
  
The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470372.png" /> takes a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470373.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470374.png" /> is a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470375.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470376.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470377.png" />-module, to the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470378.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470379.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470380.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470381.png" />. This functor takes values in the category of Abelian groups and is covariant with respect to both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470382.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470383.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470384.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470385.png" /> is a bimodule, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470386.png" /> may be equipped with an additional structure. Namely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470387.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470388.png" />-module, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470389.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470390.png" />-module, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470391.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470392.png" />-module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470393.png" /> is a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470394.png" />-module. The study of the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470395.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470396.png" />, and also of their derived functors, is one of the fundamental problems of homological algebra.
+
The functor $\tensor_A$ takes a pair $M, N$, where $M$ is a right $A$-module and $N$ is a left $A$-module, to the tensor product $M\tensor_A N$ of $M$ and $N$ over $A$. This functor takes values in the category of Abelian groups and is covariant with respect to both $M$ and $N$. When $M$ or $N$ is a bimodule, the group $M \tensor_A N$ may be equipped with an additional structure. Namely, if $M$ is a $(B, A)$-module, $M\tensor_A N$ is a $B$-module, and if $N$ is an $(A, B)$-module, then $M\tensor_A N$ is a right $B$-module. The study of the functors $\Hom$ and $\tensor$, and also of their derived functors, is one of the fundamental problems of homological algebra.
  
Many important types of modules can be characterized in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470397.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470398.png" />. Thus, a [[Projective module|projective module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470399.png" /> is defined by the requirement that the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470400.png" /> (as a functor in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470401.png" />) is exact (cf. [[Exact functor|Exact functor]]). Similarly, an [[Injective module|injective module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470402.png" /> is defined by the requirement of exactness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470403.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470404.png" />). A [[Flat module|flat module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470405.png" /> is defined by the requirement of exactness of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470406.png" />.
+
Many important types of modules can be characterized in terms of $\Hom$ and $\tensor$. Thus, a
 +
[[Projective module|projective module]] $M$ is defined by the requirement that the functor $\Hom_A(M, X)$ (as a functor in $X$) is exact (cf.
 +
[[Exact functor|Exact functor]]). Similarly, an
 +
[[Injective module|injective module]] $N$ is defined by the requirement of exactness of $\Hom_A(X, N)$ (in $X$). A
 +
[[Flat module|flat module]] $M$ is defined by the requirement of exactness of the functor $M \tensor_A X$.
  
A module over a given ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470407.png" /> can be considered from two points of view.
+
A module over a given ring $A$ can be considered from two points of view.
  
A) Modules can be studied from the point of view of their intrinsic structure. The fundamental problem here is the complete classification of modules, that is, the construction for each module of a system of invariants which characterizes the module up to an isomorphism, and, given a set of invariants, the ability to construct a module with those invariants. For certain types of rings such a description is possible. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470408.png" /> is a finitely-generated module over the group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470409.png" /> of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470410.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470411.png" /> is a field of characteristic coprime with the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470412.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470413.png" /> is representable as a finite direct sum of irreducible submodules (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470414.png" /> is completely reducible, cf. [[Completely-reducible module|Completely-reducible module]]). This representation is unique up to an isomorphism (the choice of the irreducible modules is, in general, not unique). All irreducible submodules also admit a simple description: All of them are contained in the [[Regular representation|regular representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470415.png" /> and are in one-to-one correspondence with the irreducible characters of the group. Modules over principal ideal rings and over Dedekind rings also have a simple description. Namely, any finitely-generated module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470416.png" /> over a [[Principal ideal ring|principal ideal ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470417.png" /> is isomorphic to a finite direct sum of modules of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470418.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470419.png" /> are ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470420.png" /> (possibly null), and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470421.png" />. The ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470422.png" /> are uniquely determined by this last condition. Thus, the set of invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470423.png" /> completely determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470424.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470425.png" /> is a finitely-generated module over a [[Dedekind ring|Dedekind ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470426.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470427.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470428.png" /> is a torsion module (periodic module) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470429.png" /> is a [[Torsion-free module|torsion-free module]] (the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470430.png" /> is not unique). The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470431.png" /> is annihilated by some ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470432.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470433.png" /> and, consequently, is a module over the principal ideal ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470434.png" /> and admits the description given above; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470435.png" /> is representable in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470436.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470437.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470438.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470439.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470440.png" />-fold direct sum. The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470441.png" /> is, up to an isomorphism, determined by two invariants: the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470442.png" /> and the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470443.png" /> in the ideal class group.
+
A) Modules can be studied from the point of view of their intrinsic structure. The fundamental problem here is the complete classification of modules, that is, the construction for each module of a system of invariants which characterizes the module up to an isomorphism, and, given a set of invariants, the ability to construct a module with those invariants. For certain types of rings such a description is possible. For example, if $M$ is a finitely-generated module over the group ring $KG$ of a finite group $G$, where $K$ is a field of characteristic coprime with the order of $G$, then $M$ is representable as a finite direct sum of irreducible submodules ($M$ is completely reducible, cf.
 +
[[Completely-reducible module|Completely-reducible module]]). This representation is unique up to an isomorphism (the choice of the irreducible modules is, in general, not unique). All irreducible submodules also admit a simple description: All of them are contained in the
 +
[[Regular representation|regular representation]] of $G$ and are in one-to-one correspondence with the irreducible characters of the group. Modules over principal ideal rings and over Dedekind rings also have a simple description. Namely, any finitely-generated module $M$ over a
 +
[[Principal ideal ring|principal ideal ring]] $A$ is isomorphic to a finite direct sum of modules of the form $A/\frakA_i$, where $\frakA_i$ are ideals of $A$ (possibly null), and where $\frakA_1 \subseteq \cdots \subseteq \frakA_m \ne A$. The ideals $\frakA_i$ are uniquely determined by this last condition. Thus, the set of invariants $\{\frakA_i\}$ completely determines $M$. If $M$ is a finitely-generated module over a
 +
[[Dedekind ring|Dedekind ring]] $A$, then $M = M_1 \oplus M_2$, where $M_2$ is a torsion module (periodic module) and $M_1$ is a
 +
[[Torsion-free module|torsion-free module]] (the choice of $M_1$ is not unique). The module $M_2$ is annihilated by some ideal $\frakA$ of $A$ and, consequently, is a module over the principal ideal ring $A/\frakA$ and admits the description given above; $M_1$ is representable in the form $(\bigoplus^n A) \oplus \frakB$, where $\frakB$ is an ideal of $A$ and $\bigoplus^n$ is the $n$-fold direct sum. The module $M_1$ is, up to an isomorphism, determined by two invariants: the number $n$ and the class of $\frakB$ in the ideal class group.
  
B) Another approach to the study of modules consists of studying the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470444.png" />-mod and in considering a given module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470445.png" /> as an object of this category. Such a study is the object of [[Homological algebra|homological algebra]] and [[Algebraic K-theory|algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470446.png" />-theory]]. On this route many important and deep results have been found.
+
B) Another approach to the study of modules consists of studying the category $A$-mod and in considering a given module $M$ as an object of this category. Such a study is the object of
 +
[[Homological algebra|homological algebra]] and
 +
[[Algebraic K-theory|algebraic $K$-theory]]. On this route many important and deep results have been found.
  
Often, modules which carry some extra structure are considered. Thus one considers graded modules, filtered modules, topological modules, modules with a [[Sesquilinear form|sesquilinear form]], etc. (cf. [[Graded module|Graded module]]; [[Topological module|Topological module]]; [[Filtered module|Filtered module]]).
+
Often, modules which carry some extra structure are considered. Thus one considers graded modules, filtered modules, topological modules, modules with a
 +
[[Sesquilinear form|sesquilinear form]], etc. (cf.
 +
[[Graded module|Graded module]];
 +
[[Topological module|Topological module]];
 +
[[Filtered module|Filtered module]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki,   "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Lang,   "Algebra" , Addison-Wesley (1984)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A.I. Kostrikin,   "Introduction to algebra" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> N. Jacobson,   "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> I.N. Herstein,   "Noncommutative rings" , Math. Assoc. Amer. (1968)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> C. Faith,   "Algebra: rings, modules and categories" , '''1–2''' , Springer (1981–1976)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> H. Cartan,   S. Eilenberg,   "Homological algebra" , Princeton Univ. Press (1956)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> S. MacLane,   "Homology" , Springer (1963)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> H. Bass,   "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470447.png" />-theory" , Benjamin (1968)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> J.W. Milnor,   "Introduction to algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064470/m064470448.png" />-theory" , Princeton Univ. Press (1971)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD>
 +
<TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) {{MR|0354207}} {{ZBL|}} </TD>
 +
</TR><TR><TD valign="top">[2]</TD>
 +
<TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR|0360549}} {{ZBL|0279.13001}} </TD>
 +
</TR><TR><TD valign="top">[3]</TD>
 +
<TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) {{MR|0682756}} {{ZBL|0319.17002}} </TD>
 +
</TR><TR><TD valign="top">[4]</TD>
 +
<TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1984) {{MR|0783636}} {{ZBL|0712.00001}} </TD>
 +
</TR><TR><TD valign="top">[5]</TD>
 +
<TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German) {{MR|1541390}} {{ZBL|1032.00002}} {{ZBL|1032.00001}} {{ZBL|0903.01009}} {{ZBL|0781.12003}} {{ZBL|0781.12002}} {{ZBL|0724.12002}} {{ZBL|0724.12001}} {{ZBL|0569.01001}} {{ZBL|0534.01001}} {{ZBL|0997.00502}} {{ZBL|0997.00501}} {{ZBL|0316.22001}} {{ZBL|0297.01014}} {{ZBL|0221.12001}} {{ZBL|0192.33002}} {{ZBL|0137.25403}} {{ZBL|0136.24505}} {{ZBL|0087.25903}} {{ZBL|0192.33001}} {{ZBL|0067.00502}} </TD>
 +
</TR><TR><TD valign="top">[6]</TD>
 +
<TD valign="top"> A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian) {{MR|0661256}} {{ZBL|0482.00001}} </TD>
 +
</TR><TR><TD valign="top">[7]</TD>
 +
<TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) {{MR|0081264}} {{ZBL|0073.02002}} </TD>
 +
</TR><TR><TD valign="top">[8]</TD>
 +
<TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) {{MR|1535024}} {{MR|0227205}} {{ZBL|0177.05801}} </TD>
 +
</TR><TR><TD valign="top">[9]</TD>
 +
<TD valign="top"> C. Faith, "Algebra: rings, modules and categories" , '''1–2''' , Springer (1981–1976) {{MR|0551052}} {{MR|0491784}} {{MR|0366960}} {{ZBL|0508.16001}} {{ZBL|0266.16001}} </TD>
 +
</TR><TR><TD valign="top">[10]</TD>
 +
<TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) {{MR|0077480}} {{ZBL|0075.24305}} </TD>
 +
</TR><TR><TD valign="top">[11]</TD>
 +
<TD valign="top"> S. MacLane, "Homology" , Springer (1963) {{MR|}} {{ZBL|0818.18001}} {{ZBL|0328.18009}} </TD>
 +
</TR><TR><TD valign="top">[12]</TD>
 +
<TD valign="top"> H. Bass, "Algebraic $K$-theory" , Benjamin (1968) {{MR|249491}} {{ZBL|}} </TD>
 +
</TR><TR><TD valign="top">[13]</TD>
 +
<TD valign="top"> J.W. Milnor, "Introduction to algebraic $K$-theory" , Princeton Univ. Press (1971) {{MR|349811}} {{ZBL|}} </TD>
 +
</TR></table>

Latest revision as of 06:31, 23 July 2018

$\newcommand{\tensor}{\otimes}$ $\newcommand{\frakA}{\mathfrak{A}}$ $\newcommand{\frakB}{\mathfrak{B}}$ $\newcommand{\lieg}{\mathfrak{g}}$ $\newcommand{\too}{\longrightarrow}$ $\newcommand{\inv}{^{-1}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\DeclareMathOperator{\Mod}{Mod}$ $\DeclareMathOperator{\Ker}{Ker}$ $\DeclareMathOperator{\Coker}{Coker}$ $\DeclareMathOperator{\Coim}{Coim}$

An Abelian group with the distributive action of a ring. A module is a generalization of a (linear) vector space over a field $K$, when $K$ is replaced by a ring.

Let a ring $A$ be given. An additive Abelian group $M$ is called a left $A$-module if there is a mapping $A\times M \to M$ whose value on a pair $(a, m)$, for $a \in A$, $m \in M$, written $am$, satisfies the axioms:

1) $a(m_1 + m_2) = am_1 + am_2$;

2) $(a_1 + a_2)m = a_1 m + a_2 m$;

3) $a_1(a_2 m) = (a_1 a_2) m$. If $A$ is a ring with identity, then it is usual to require in addition that for any $m \in M$, $1m = m$. A module with this property is called unitary or unital (cf. Unitary module).

Right $A$-modules are defined similarly; axiom 3) is replaced by $(ma_1)a_2 = m(a_1 a_2)$. Any right $A$-module can be considered as a left $A^\text{opp}$-module over the opposite ring $A^\text{opp}$ anti-isomorphic to $A$; hence, corresponding to any result about right $A$-modules there is a result about left $A^\text{opp}$-modules, and conversely. When $A$ is commutative, any left $A$-module can be considered as a right $A$-module and the distinction between left and right modules disappears. Below only left $A$-modules are discussed.

The simplest examples of modules (finite Abelian groups; they are $\ZZ$-modules) were known already to C.F. Gauss as class groups of binary quadratic forms. The general notion of a module was first encountered in the 1860's till 1880's in the work of R. Dedekind and L. Kronecker, devoted to the arithmetic of algebraic number and function fields. At approximately the same time research on finite-dimensional associative algebras, in particular, group algebras of finite groups (B. Pierce, F. Frobenius), led to the study of ideals of certain non-commutative rings. At first the theory of modules was developed primarily as a theory of ideals of a ring. Only later, in the work of E. Noether and W. Krull, it was observed that it was more convenient to formulate and prove many results in terms of arbitrary modules, and not just ideals. Subsequent developments of the theory of modules were connected with the application of methods and ideas of the theory of categories (cf. Category), in particular, methods of homological algebra.

Examples of modules.

1) Any Abelian group $M$ is a module over the ring of integers $\ZZ$. For $a \in \ZZ$ and $m \in M$ the product $am$ is defined as the result of adding $m$ to itself $a$ times.

2) When $A$ is a field, the notion of a unitary $A$-module is exactly equivalent to the notion of a linear vector space over $A$.

3) An $n$-dimensional vector space $V$ over a field $K$ (provided with coordinates) can be considered as a module over the ring $M_n(K)$ of all $(n\times n)$-matrices with coefficients in $K$. For $v \in V$ and $X \in M_n(K)$ the product $Xv$ is defined as multiplication of the matrix $X$ by the column of coordinates of the vector $v$.

4) An associative ring (cf. Associative rings and algebras) $A$ is a left $A$-module. Multiplication of elements of the ring by elements of the module is ordinary multiplication in $A$.

5) The set of differential forms on a smooth manifold $X$ has the natural structure of a module over the ring of all smooth functions on $X$.

6) Connected with any Abelian group $M$ is the associative ring with identity, $\End(M)$, of all endomorphisms of $M$. The group $M$ has a natural $\End(M)$-module structure.

If there is an $A$-module structure on $M$, for some ring $A$, then the mapping $m \mapsto am$ is an endomorphism of $M$ for any $a \in A$. Associating with the element $a \in A$ the endomorphism of $M$ that it generates, one obtains a homomorphism $\phi$ of $A$ into $\End(M)$. Conversely, any homomorphism $\phi: A \to \End(M)$ defines the structure of an $A$-module on $M$. Thus, the specification of an $A$-module structure on an Abelian group $M$ is equivalent to the specification of a homomorphism of rings $\phi: A \to \End(M)$. Such a homomorphism is also called a representation of the ring $A$, and $M$ is called a representation module. Connected with any representation $\phi$ is a two-sided ideal $\Ann(M) = \Ker \phi$, consisting of the $a \in A$ such that $am = 0$ for all $m \in M$. This ideal is called the annihilator of the module $M$. When $\Ann(M) = 0$, the representation is called faithful and $M$ is called a faithful module (or faithful representation).

It is obvious that a module $M$ can also be considered as a module over the quotient ring $A/\Ann(M)$. In particular, although the definition of a module does not assume the associativity of $A$, the ring $A/\Ann(M)$ is always associative. Therefore, in the majority of cases the discussion may be restricted to modules over associative rings. Everywhere below, unless stated otherwise, $A$ is assumed to be associative.

$G$-modules.

Let $G$ be a group. An additive Abelian group $M$ is called a left $G$-module if there is a mapping $G\times M \to M$ whose value at a pair $(g, m)$, where $g \in G$, $m \in M$, is written as $gm$, and where for any $g \in G$ the mapping $m \mapsto gm$ is an endomorphism of $M$; for any $g_1, g_2 \in G$, $m \in M$, $(g_1 g_2)m = g_1(g_2 m)$; and for all $m \in M$, $1m = m$, where 1 is the identity of $G$. For any $g \in G$ the mapping $m \mapsto gm$ is an automorphism of the group $M$.

Right $G$-modules may be defined similarly.

Examples of $K$-modules.

1) Let $k$ be a Galois extension of a field $G$ with Galois group $K$. Then the additive and multiplicative groups of $K$ have the natural structure of $G$-modules. If $k$ is an algebraic number field, then other $G$-modules are: the additive group of the ring of integers of $K$, the group of units of $K$, the group of divisors and the divisor class group of $K$, etc. A module over a Galois group is called a Galois module.

2) Let an extension of an Abelian group $M$ be given, that is, an exact sequence of groups

$$1 \too M \too F \too G \too 1,$$ where $M$ is an Abelian normal subgroup of $F$ and $G$ is an arbitrary group. Then $M$ can be given the natural structure of a $G$-module by putting, for $g \in G$, $m \in M$, $gm = \overline g m \overline g\inv$, where $\overline g$ is an inverse image of $g$ in $F$.

When the group operation in the Abelian group $M$ is written multiplicatively (for example, if $M$ is the multiplicative group of a field), the notation $m^g$ is also used instead of $gm$, that is, the action of $G$ is written exponentially.

Let a $G$-module $M$ be given. By associating with an element $g \in G$ the automorphism $m \mapsto gm$ of $M$, a homomorphism of $G$ into the group of invertible elements of the ring $\End(M)$ is obtained. Conversely, any homomorphims of $G$ into the group of invertible elements of $\End(M)$ gives $M$ the structure of a $G$-module.

The notions of a module over a ring and a $G$-module are closely connected. Namely, any $G$-module $M$ can be regarded as a module over the group ring (cf. Group algebra) $\ZZ G$ if the action of $G$ on $M$ is extended linearly, that is, if one puts

$$\left(\sum a_i g_i\right) m = \sum a_i (g_i m),$$ where $a_i \in \ZZ$, $g_i \in G$, $m \in M$. Conversely, given a unitary $\ZZ G$-module structure on $M$, $M$ may be regarded to be a $G$-module.

When $M$ is simultaneously a $K$-module over a commutative ring $K$ and a $G$-module, where the action of the elements of $G$ on $M$ commutes with the action of the elements of $K$, then $M$ may be given the structure of a $KG$-module by linearly extending the action from $G$ to $KG$. For example, if $V$ is a linear vector space over a field $K$, then the specification of a $KG$-module structure on $V$ is equivalent to giving a representation of $G$ in $V$.

Using the standard involution $g \mapsto g\inv$ in $G$, any left $G$-module $M$ can be made into a right $G$-module by putting $mg = g\inv m$ for $m \in M$, $g \in G$. Similarly, any right $G$-module can be made into a left $G$-module.

Modules over a Lie algebra.

Let $\lieg$ be a Lie algebra over a commutative ring $K$ and let $M$ be a $K$-module. The specification of a $\lieg$-module structure on $M$ consists of the specification of a $K$-endomorphism $m \mapsto gm$ of the group $M$ for each $g \in \lieg$, where the axiom

$$[g_1, g_2] m = g_1(g_2 m) - g_2(g_1 m)$$ holds for $g_1, g_2 \in \lieg$, $m \in M$. This definition differs from that of an $A$-module given earlier. Giving a $\lieg$-module structure on $M$ is equivalent to giving a Lie algebra homomorphism of $\lieg$ into the Lie algebra of the ring $\End(M)$. A module over a Lie algebra $\lieg$ may also be regarded as a module in the usual sense over the universal enveloping algebra of $\lieg$.

Constructions in the theory of modules.

Starting from a given $A$-module it is possible to obtain new $A$-modules by a number of standard constructions. Thus, with any module $M$ is associated the lattice of its submodules. For example, if $A$ is considered as left module over itself, then its left submodules are precisely the left ideals in $A$. A number of important types of modules are defined in terms of the lattice of submodules. For example, the condition for termination of a descending (ascending) chain of submodules defines Artinian modules (respectively, Noetherian modules, cf. Artinian module; Noetherian module). The condition for absence of non-trivial submodules, that is, submodules other than 0 or $M$, defines irreducible or simple modules (cf. Irreducible module).

For a module $M$ and any submodule $N$, the quotient group $M/N$ can be given the structure of an $A$-module. This module is called the quotient module of $M$ over $N$.

A homomorphism of $A$-modules is defined as an Abelian group homomorphism $f : M \to N$ commuting with multiplication by elements of $A$, that is, $f(am) = af(m)$ for all $m \in M$, $a \in A$. If two homomorphisms $f_1, f_2 : M \to N$ are given, then their sum, defined by $(f_1 + f_2)(m) = f_1(m) = f_2(m)$, is again a homomorphism of $A$-modules. This addition gives an Abelian group structure to the set $\Hom_A(M, N)$ of all homomorphisms of $M$ into $N$. For any homomorphism $f : M \to N$ the submodules $\Ker f$ (the kernel of $f$) and $\Im f$ (the image of $f$), and also the quotient modules $\Coker f = N / \Im f$ (the cokernel of $f$) and $\Coim f = M/\Ker f$ (the coimage of $f$) are defined. The modules $\Im f$ and $\Coim f$ are canonically isomorphic and therefore usually identified. For example, for any left ideal $J$ of $A$ the quotient module $A/J$ is defined. The module $A/J$ is irreducible if and only if $J$ is a maximal left ideal (cf. Maximal ideal). If $M$ is an irreducible $A$-module not annihilating the ring $A$, then $M$ is isomorphic to $A/J$ for some maximal left ideal $J$.

For any family of $A$-modules $\{M_i\}$, where $i$ runs through some index set $J$, the direct sum and direct product of $\{M_i\}$ exist in the category of $A$-modules. Here an element of the direct product $\prod_{i \in J} M_i$ may be interpreted as a vector $(ldots, m_i, \ldots)$ the components of which are indexed by $J$ and where for each $i$, $m_i \in M_i$. The sum of such vectors and their multiplication by elements of the ring are defined componentwise. The direct sum $\sum_{i \in J} M_i$ of the family $\{M_i\}$ can be interpreted as the submodule of the direct product consisting of the vectors all components of which, except for finitely many, are equal to zero.

For a projective (inductive) system of $A$-modules the projective (inductive) limit of this system can be naturally equipped with the structure of an $A$-module. The direct product and direct sum may be considered as special cases of the notions of a projective and an inductive limit.

Generators and relations.

Let $X$ be a subset of an $A$-module $M$. The submodule generated by $X$ is the intersection of the submodules of $M$ which contain $X$. If this submodule coincides with $M$, then $X$ is called a family (system) of generators of the module $M$. A module admitting a finite family of generators is called a finitely-generated module. For example, in a Noetherian ring any ideal is a finitely-generated module. A direct sum of a finite number of finitely-generated modules is again finitely generated. Any quotient module of a finitely-generated module is also finitely generated. For the construction of a system of generators for a module $M$, the Nakayama lemma often turns out to be useful: For any ideal $\frakA$ contained in the radical of a ring $A$ the condition $\frakA M = M$ implies $M = 0$. In particular, under the conditions of Nakayama's lemma elements $m_1, \ldots, m_r$ form a system of generators for $M$ if their images generate the module $M/\frakA M$. This is used particularly often when $A$ is a local ring and $\frakA$ is the maximal ideal in $A$.

Let $M$ be a module with system of generators $\{x_i\}_{i \in J}$. Then a mapping $\phi : y_i \to x_i$ defines an epimorphism of the free $A$-module $F$ with generators $\{y_i\}_{i \in I}$ onto $M$ ($F$ can be defined as the set of formal finite sums $\sum a_i y_i$, $a_i \in A$, and $\phi$ is extended from the generators to $F$ by linearity). The elements of $R = \Ker \phi$ are called relations between the generators $\{x_i\}$ of $M$. If $M$ can be represented as a quotient module of a finitely-generated free module $F$ so that the module of relations $R$ is also finitely generated, then $M$ is called a finitely-presented module. For example, over a Noetherian ring $A$ any finitely-generated module is finitely presented. In general, being finitely generated does not imply being finitely presented.

Change of rings.

There are standard constructions which allow an $A$-module $M$ to be considered as a module over some other ring. For example, let $\phi : B \to A$ be a homomorphism of rings. Then, putting $bm = \phi(b) m$, $M$ can be considered as a $B$-module. The resulting $B$-module is said to be obtained by base change or, in particular in the case that $B$ is a subring of $A$, by restriction of scalars. If $M$ is a unitary $A$-module and $\phi$ takes the identity to the identity, $M$ becomes a unitary $B$-module.

Let a ring homomorphism $\phi: A \to B$ and an $A$-module $M$ be given. Then $B$ may be given the structure of a $(B, A)$-module (cf. Bimodule) by putting $ba = b\phi(a)$ for $b\in B$, $a \in A$, and the left $B$-module $B \tensor_A M$ can be considered. One says that this module is obtained from $M$ by extension of scalars.

The category of modules.

The class of all modules over a given ring $A$ with homomorphisms of modules as morphisms forms an Abelian category, denoted, for instance, by $A$-mod or $\Mod_A$. The most important functors defined on this category are $\Hom$ (homomorphism) and $\tensor$ (tensor product). The functor $\Hom$ takes values in the category of Abelian groups and associates to a pair of $A$-modules $M, N$ the group $\Hom_A(M, N)$. For $f : M_1 \to M$ and $\phi : N \to N_1$ the mappings

$$f' : \Hom_A(M, N) \to \Hom_A(M_1, N)$$ and

$$\phi' : \Hom_A(M, N) \to \Hom_A(M, N_1)$$ are defined in the obvious way; that is, the functor $\Hom$ is contravariant in its first argument and covariant in the second. When $M$ or $N$ carry a bimodule structure, the group $\Hom_A(M,N)$ has an additional module structure. If $N$ is an $(A, B)$-module, $\Hom_A(M,N)$ is a right $B$-module and if $M$ is an $(A,B)$-module, then $\Hom_A(M, N)$ is a left $B$-module.

The functor $\tensor_A$ takes a pair $M, N$, where $M$ is a right $A$-module and $N$ is a left $A$-module, to the tensor product $M\tensor_A N$ of $M$ and $N$ over $A$. This functor takes values in the category of Abelian groups and is covariant with respect to both $M$ and $N$. When $M$ or $N$ is a bimodule, the group $M \tensor_A N$ may be equipped with an additional structure. Namely, if $M$ is a $(B, A)$-module, $M\tensor_A N$ is a $B$-module, and if $N$ is an $(A, B)$-module, then $M\tensor_A N$ is a right $B$-module. The study of the functors $\Hom$ and $\tensor$, and also of their derived functors, is one of the fundamental problems of homological algebra.

Many important types of modules can be characterized in terms of $\Hom$ and $\tensor$. Thus, a projective module $M$ is defined by the requirement that the functor $\Hom_A(M, X)$ (as a functor in $X$) is exact (cf. Exact functor). Similarly, an injective module $N$ is defined by the requirement of exactness of $\Hom_A(X, N)$ (in $X$). A flat module $M$ is defined by the requirement of exactness of the functor $M \tensor_A X$.

A module over a given ring $A$ can be considered from two points of view.

A) Modules can be studied from the point of view of their intrinsic structure. The fundamental problem here is the complete classification of modules, that is, the construction for each module of a system of invariants which characterizes the module up to an isomorphism, and, given a set of invariants, the ability to construct a module with those invariants. For certain types of rings such a description is possible. For example, if $M$ is a finitely-generated module over the group ring $KG$ of a finite group $G$, where $K$ is a field of characteristic coprime with the order of $G$, then $M$ is representable as a finite direct sum of irreducible submodules ($M$ is completely reducible, cf. Completely-reducible module). This representation is unique up to an isomorphism (the choice of the irreducible modules is, in general, not unique). All irreducible submodules also admit a simple description: All of them are contained in the regular representation of $G$ and are in one-to-one correspondence with the irreducible characters of the group. Modules over principal ideal rings and over Dedekind rings also have a simple description. Namely, any finitely-generated module $M$ over a principal ideal ring $A$ is isomorphic to a finite direct sum of modules of the form $A/\frakA_i$, where $\frakA_i$ are ideals of $A$ (possibly null), and where $\frakA_1 \subseteq \cdots \subseteq \frakA_m \ne A$. The ideals $\frakA_i$ are uniquely determined by this last condition. Thus, the set of invariants $\{\frakA_i\}$ completely determines $M$. If $M$ is a finitely-generated module over a Dedekind ring $A$, then $M = M_1 \oplus M_2$, where $M_2$ is a torsion module (periodic module) and $M_1$ is a torsion-free module (the choice of $M_1$ is not unique). The module $M_2$ is annihilated by some ideal $\frakA$ of $A$ and, consequently, is a module over the principal ideal ring $A/\frakA$ and admits the description given above; $M_1$ is representable in the form $(\bigoplus^n A) \oplus \frakB$, where $\frakB$ is an ideal of $A$ and $\bigoplus^n$ is the $n$-fold direct sum. The module $M_1$ is, up to an isomorphism, determined by two invariants: the number $n$ and the class of $\frakB$ in the ideal class group.

B) Another approach to the study of modules consists of studying the category $A$-mod and in considering a given module $M$ as an object of this category. Such a study is the object of homological algebra and algebraic $K$-theory. On this route many important and deep results have been found.

Often, modules which carry some extra structure are considered. Thus one considers graded modules, filtered modules, topological modules, modules with a sesquilinear form, etc. (cf. Graded module; Topological module; Filtered module).

References

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How to Cite This Entry:
Module. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Module&oldid=17178
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article