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| − | A non-zero [[Unitary module|unitary module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i0526101.png" /> over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i0526102.png" /> with a unit element that contains only two submodules: the null module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i0526103.png" /> itself. | + | A non-zero [[unital module]] $M$ over a [[unital ring]] $R$ that contains only two submodules: the null module and $M$ itself. |
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| − | Examples. 1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i0526104.png" /> is the ring of integers, then the irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i0526105.png" />-modules are the Abelian groups of prime order. 2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i0526106.png" /> is a skew-field, then the irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i0526107.png" />-modules are the one-dimensional vector spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i0526108.png" />. 3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i0526109.png" /> is a skew-field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261010.png" /> is a left vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261012.png" /> is the ring of linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261013.png" /> (or a dense subring of it), then the right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261014.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261015.png" /> is irreducible. 4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261016.png" /> is a group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261017.png" /> is a field, then the irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261018.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261019.png" /> are precisely the irreducible modules over the [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261020.png" />. | + | Examples. 1) If $R = \mathbf{Z}$ is the ring of integers, then the irreducible $R$-modules are the Abelian groups of prime order. 2) If $R$ is a [[skew-field]], then the irreducible $R$-modules are the one-dimensional vector spaces over $R$. 3) If $D$ is a skew-field, $V$ is a left vector space over $D$ and $R = \End_D(V)$ is the ring of linear transformations of $V$ (or a dense subring of it), then the right $R$-module $V$ is irreducible. 4) If $G$ is a group and $k$ is a field, then the [[irreducible representation]]s of $G$ over $K$ are precisely the irreducible modules over the [[group algebra]] $R = k[G]$. |
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| − | A right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261021.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261022.png" /> is irreducible if and only if it is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261024.png" /> is a maximal right ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261027.png" /> are irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261028.png" />-modules and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261029.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261030.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261031.png" /> is an isomorphism (which implies that the endomorphism ring of an irreducible module is a skew-field). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261032.png" /> is an algebra over an algebraically closed field and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261034.png" /> are irreducible modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052610/i05261035.png" />, then (Schur's lemma) | + | A right $R$-module $M$ is irreducible if and only if it is isomorphic to $R/I$, where $I$ is a maximal right ideal in $R$. If $A$ and $B$ are irreducible $R$-modules and $f \in \Hom_R(A,B)$, then either $f=0$ or $f$ is an isomorphism (which implies that the endomorphism ring of an irreducible module is a skew-field). If $R$ is an algebra over an algebraically closed field $K$ and if $A$ and $B$ are irreducible modules over $R$, then ([[Schur lemma|Schur's lemma]]) |
| | + | $$ |
| | + | \Hom_R(A,B) = \begin{cases} K & \ \text{if}\ A \cong B\ ; \\ 0 & \ \text{otherwise} \ .\end{cases} |
| | + | $$ |
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| − | <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/i/i052/i052610/i05261036.png" /></td> </tr></table>
| + | The concept of an irreducible module is fundamental in the theories of rings and group representations. By means of it one defines the [[composition sequence]] and the [[socle]] of a module, the [[Jacobson radical]] of a module and of a ring, and a [[completely-reducible module]]. Irreducible modules are involved in the definition of a number of important classes of rings: [[classical semi-simple ring]]s, [[primitive ring]]s, and others. |
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| − | The concept of an irreducible module is fundamental in the theories of rings and group representations. By means of it one defines the [[Composition sequence|composition sequence]] and the [[Socle|socle]] of a module, the [[Jacobson radical|Jacobson radical]] of a module and of a ring, and a [[Completely-reducible module|completely-reducible module]]. Irreducible modules are involved in the definition of a number of important classes of rings: classical semi-simple rings, primitive rings, and others (cf. [[Classical semi-simple ring|Classical semi-simple ring]]; [[Primitive ring|Primitive ring]]).
| + | ====References==== |
| | + | <table> |
| | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR> |
| | + | <TR><TD valign="top">[2]</TD> <TD valign="top"> C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)</TD></TR> |
| | + | <TR><TD valign="top">[3]</TD> <TD valign="top"> J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)</TD></TR> |
| | + | <TR><TD valign="top">[4]</TD> <TD valign="top"> C. Faith, "Algebra: rings, modules, and categories" , '''1–2''' , Springer (1973–1976)</TD></TR> |
| | + | </table> |
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| − | ====References====
| + | {{TEX|done}} |
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> C. Faith, "Algebra: rings, modules, and categories" , '''1–2''' , Springer (1973–1976)</TD></TR></table>
| |
simple module
A non-zero unital module $M$ over a unital ring $R$ that contains only two submodules: the null module and $M$ itself.
Examples. 1) If $R = \mathbf{Z}$ is the ring of integers, then the irreducible $R$-modules are the Abelian groups of prime order. 2) If $R$ is a skew-field, then the irreducible $R$-modules are the one-dimensional vector spaces over $R$. 3) If $D$ is a skew-field, $V$ is a left vector space over $D$ and $R = \End_D(V)$ is the ring of linear transformations of $V$ (or a dense subring of it), then the right $R$-module $V$ is irreducible. 4) If $G$ is a group and $k$ is a field, then the irreducible representations of $G$ over $K$ are precisely the irreducible modules over the group algebra $R = k[G]$.
A right $R$-module $M$ is irreducible if and only if it is isomorphic to $R/I$, where $I$ is a maximal right ideal in $R$. If $A$ and $B$ are irreducible $R$-modules and $f \in \Hom_R(A,B)$, then either $f=0$ or $f$ is an isomorphism (which implies that the endomorphism ring of an irreducible module is a skew-field). If $R$ is an algebra over an algebraically closed field $K$ and if $A$ and $B$ are irreducible modules over $R$, then (Schur's lemma)
$$
\Hom_R(A,B) = \begin{cases} K & \ \text{if}\ A \cong B\ ; \\ 0 & \ \text{otherwise} \ .\end{cases}
$$
The concept of an irreducible module is fundamental in the theories of rings and group representations. By means of it one defines the composition sequence and the socle of a module, the Jacobson radical of a module and of a ring, and a completely-reducible module. Irreducible modules are involved in the definition of a number of important classes of rings: classical semi-simple rings, primitive rings, and others.
References
| [1] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |
| [2] | C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) |
| [3] | J. Lambek, "Lectures on rings and modules" , Blaisdell (1966) |
| [4] | C. Faith, "Algebra: rings, modules, and categories" , 1–2 , Springer (1973–1976) |