of functions, vectors, matrices, tensors, etc.) over it (more generally, a module $ M $).
Given a left module $ M $
2 KB (358 words) - 08:12, 6 June 2020
A property $P$ of a [[commutative ring]] $A$ or an $A$-module $M$ that is true for $A$ (or $M$) if and only if a similar property holds f
However, the property of an $A$-module $M$ of being [[Free module|free]] is not local.
2 KB (320 words) - 08:07, 1 January 2017
...) rank of an [[Abelian group|Abelian group]] (which can be considered as a module over the ring $\mathbf Z$). In the non-Abelian case two concepts of the ran
...<TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra I" , Addison-Wesley (1974) pp. Chapt. II (Translated from French)
4 KB (631 words) - 10:02, 15 April 2014
...q 0$ for every submodule $S \subset E$, $S \neq 0$. They proved that every module has an injective envelope and that if $M \subset E _ { 1 }$ and $M \subset
...lat left $R$-module is projective (cf. also [[Projective module|Projective module]]). Projective covers (when they exist) are also unique up to isomorphism.
6 KB (1,031 words) - 16:55, 1 July 2020
# if $E$ is a [[Free module|free $A$-module]] with basis $e_1,\ldots,e_{2n}$ and if $$
|align="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics", '''2. Linear and multilinear algebra''', Addison-Wesley (1973) pp. Chapt.
1 KB (251 words) - 19:57, 30 November 2014
module of finite type; or 2) there exists a faithful $ A [ x] $-
module that is an $ A $-
3 KB (562 words) - 22:12, 5 June 2020
...th coefficients from $\Phi$. In other words, this free algebra is a [[free module]] over $\Phi$ with the above-mentioned magma as its base. When $\Phi$ is th
...orial methods. Free groups, polynomials, and free algebras'', CMS Books in Mathematics '''19''' Springer (2004) {{ISBN|0-387-40562-3}} {{ZBL|1039.16024}}
1 KB (163 words) - 08:20, 12 November 2023
The endomorphism ring of a [[tilting module]] over a finite-dimensional hereditary algebra (cf. also [[Algebra]]; [[End
...ted cycles. A finite-dimensional $H$-module ${}_{H} T$ is called a tilting module if
8 KB (1,215 words) - 19:50, 24 December 2023
.../legacyimages/c/c025/c025970/c02597059.png" />-contravariant tensor of the
module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Elements of
mathematics. Algebra: Algebraic structures. Multilinear algebra" , Addison-Wesley (197
10 KB (1,407 words) - 17:29, 7 February 2011
...hfrak{g}) $ is the [[Tensor algebra|tensor algebra]] of the $ \mathbb{k} $-module $ \mathfrak{g} $, $ I $ is the two-sided ideal generated by all elements of
...athfrak{g}) $ is left- and right-Noetherian. If $ \mathfrak{g} $ is a free module over an integral domain $ \mathbb{k} $, then $ U(\mathfrak{g}) $ has no zer
6 KB (970 words) - 18:59, 5 April 2023
...duct $V\times W$ of a left unitary $A$-module $V$ and of right unitary $B$-module $W$ into an $(A,B)$-[[Bimodule|bimodule]] $H$, satisfying the conditions
...es a canonical isomorphism between the $A$-module $L_2(V,W,H)$ and the $A$-module $L(V\otimes W,H)$ of all linear mappings from $V\otimes W$ into $H$.
4 KB (832 words) - 17:52, 8 December 2014
...product of an infinite number of free modules is not free. A torsion-free module of countable rank over a complete discretely-normed ring is a direct sum of
...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
5 KB (800 words) - 19:36, 5 June 2020
...an $(A,A)$-bimodule. If $V=W$, one says that $f$ is a bilinear form on the module $V$, and also that $V$ has a metric structure given by $f$. Definitions inv
Let $A$ be a commutative ring. A bilinear form on an $A$-module $V$ is said to be symmetric (or anti-symmetric or skew-symmetric) if for al
6 KB (1,157 words) - 08:58, 9 December 2016
module. A derivation in $ R $
module on the set of all derivations in $ R $
6 KB (945 words) - 17:32, 5 June 2020
...a on $X$ is obtained from the vector space $V = KX$ and forming the tensor module
...orial methods. Free groups, polynomials, and free algebras'', CMS Books in Mathematics '''19''' Springer (2004) {{ISBN|0-387-40562-3}} {{ZBL|1039.16024}}
1 KB (217 words) - 15:05, 19 November 2023
of a free module $ M $
is a free module of rank $ k \leq n $
5 KB (880 words) - 19:00, 9 January 2024
[[Abelian group]] with the distributive action of a ring. A module is a generalization of a (linear)
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
23 KB (3,918 words) - 04:31, 23 July 2018
of a module $ M $
is defined as the quotient module of the $ r $-
7 KB (1,013 words) - 19:38, 5 June 2020
on a left $R$-module $X$'' is
[[sesquilinear form]] on $X$. The module $X$ itself is then called a Hermitian space. By analogy with what is done f
5 KB (831 words) - 17:13, 9 October 2016
...me ideal $\mathfrak{p}$ of height $\ge 2$ the depth (cf. also [[Depth of a module]]) is also $\ge 2$. (Cf. [[#References|[a3]]], p. 125.)
<TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
2 KB (305 words) - 16:09, 11 September 2016
...n called linear groups. In particular, if $M$ is a free finite-dimensional module over the ring $\mathbf Z$ of integers, one speaks of crystallographic group
...ms" A.D. Aleksandrov (ed.) A.N. Kolmogorov (ed.) M.A. Lavrent'ev (ed.) , ''Mathematics, its content, methods and meaning'' , '''3''' , Amer. Math. Soc. (1962) pp.
2 KB (330 words) - 09:46, 11 August 2014
...References|[1]]], [[#References|[3]]]). By relating to the elements of the module, called vectors, new elements, called points, related to the vectors by the
...seudo-Euclidean) metric can be given. To do this one defines in a [[unital module]] a scalar vector product $ ( \mathbf a , \mathbf b ) $,
8 KB (1,172 words) - 16:55, 7 June 2020
...groups it is the trivial group, in the category of modules it is the zero module, etc. Not every category contains a null object, but a null object can alwa
* S. MacLane, "Categories for the working mathematician" Graduate Texts in Mathematics '''5''', Springer (1971) {{ISBN|0-387-98403-8}}
2 KB (327 words) - 18:38, 13 November 2023
module $ A [ S ^ {-1} ] $
module $ M $
4 KB (713 words) - 21:35, 4 January 2021
...and $V_2$ over an associative commutative ring $A$ with a unit is the $A$-module $V_1 \tensor_A V_2$ together with an $A$-bilinear mapping
...inear mapping $\beta: V_1 \times V_2 \to W$, where $W$ is an arbitrary $A$-module, there is a unique $A$-linear mapping $b : V_1 \tensor_A V_2 \to W$ such th
11 KB (1,992 words) - 03:52, 23 July 2018
...$\Gamma ^ { \operatorname{op} }$. Hence $D ( T )$ is a cotilting $\Gamma$-module.
...( \mathcal{T} , C ) = 0 \}$. Dually, there is associated with a cotilting module $T$ the subcategory $\mathcal{Y} = \operatorname { Sub } T$ of $\Lambda$-mo
16 KB (2,221 words) - 09:47, 11 November 2023
...tation of an [[ideal]] $I$ of a [[ring]] $R$ (or of a submodule $N$ of a [[module]] $M$) as an intersection of primary ideals (primary submodules, cf. [[Prim
<TR><TD valign="top">[5]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
2 KB (386 words) - 20:06, 5 October 2017
...n arbitrary module $ A $ can be represented as a quotient module of a free module $ F_{0} $, after which a similar representation $ F_{1} $ is considered for
...s $ P_{i} $ are projective, is called a '''projective resolution''' of the module $ A $. The application of a covariant additive functor $ T $ yields a compl
12 KB (1,885 words) - 23:48, 23 April 2017
...e overlapping shuffle algebra $\mathrm{OSh}(\mathbf{N})$. As a $\mathbf Z$-module, $\mathrm{OSh}(\mathbf{N})$ is free with basis $\mathbf{N}^*$, the free mon
...bra of Quasi-Symmetric Functions is Free over the Integers", ''Advances in Mathematics'' '''164''' (2001) 283–300 {{DOI|10.1006/aima.2001.2017}}</TD></TR>
4 KB (709 words) - 13:46, 20 March 2023
cf. [[Lie algebra, free|Lie algebra, free]]) is freely generated (as a module over a commutative ring $ K $)
...may then be phrased as follows: One has the direct sum decomposition (as a module over a commutative ring $ K $):
5 KB (722 words) - 18:41, 5 October 2023
geometry itself and in other fields of mathematics, particularly in
[[Tate module|Tate module]] of an Abelian variety. It is the
8 KB (1,216 words) - 20:39, 5 March 2012
...ard procedure, certain further requirements may have to be considered: the module principle, programming discipline, auto-documentation, filtration of input
3 KB (372 words) - 17:12, 7 February 2011
The Alexander module is a generalization of the group of $n$-colourings.
...ry" , Ginn (1963) {{ZBL|0126.39105}}. 4th corr. print. Graduate Texts in Mathematics '''57''', Springer (1977) {{ISBN|3-540-90272-4}} {{ZBL|0362.55001}}. Repr.
2 KB (367 words) - 18:07, 14 November 2023
module and $ f $
...athematics. Algebra: Algebraic structures. Linear algebra" , ''Elements of mathematics'' , '''1''' , Addison-Wesley (1974) pp. Chapts. 1–2 (Translated from Fren
4 KB (558 words) - 16:18, 6 June 2020
...gebra $\g$ is the linear representation $\ad$ of the algebra $\g$ into the module $\g$ acting by the formula
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) pp. Chapt. 2; 3 (Tr
3 KB (407 words) - 16:57, 27 March 2012
...and automata]]). In [[Mathematical logic|mathematical logic]] (i.e., meta-mathematics) one builds several classes of formal languages, of which first-order logic
...]). Equational logic generalizes in a similar way. For example, a [[Module|module]] over a ring is a two-sorted algebra with two universes, an Abelian group
9 KB (1,433 words) - 17:00, 1 July 2020
As a module over the [[Steenrod algebra|Steenrod algebra]],
...plex cobordism and stable homotopy groups of spheres" , ''Pure and Applied Mathematics'' , '''121''' , Acad. Press (1986)</TD></TR><TR><TD valign="top">[a7]</TD>
4 KB (633 words) - 08:47, 26 March 2023
module $ M $,
see [[Depth of a module|Depth of a module]]); the condition $ \mathop{\rm depth} A = \mathop{\rm dim} A $
9 KB (1,456 words) - 22:17, 5 June 2020
Let $A$ be an arbitrary module over the ring $R = k[x_1,\ldots,x_n]$. Then there are free $R$-modules $X_0
A finitely-generated projective module over a ring of polynomials in a finite number of variables with coefficient
6 KB (1,011 words) - 06:01, 30 September 2023
...|dynamic programming]], discrete-event systems, computer science, discrete mathematics, [[Mathematical logic|mathematical logic]], etc. (see, e.g., [[#References|
is called an idempotent semi-module over an idempotent semi-ring $ A $(
18 KB (2,598 words) - 22:11, 5 June 2020
...ly determines $L$ and is called its rank. A free Lie algebra is a free $R$-module (for bases of it see [[Basic commutator]]). A subalgebra $M$ of a free Lie
<TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from Fr
3 KB (564 words) - 19:53, 15 March 2023
module $ R ^ {q} f _ {*} ( F ) _ {y} $
module.
9 KB (1,267 words) - 08:08, 6 June 2020
...$A$, considered as a right $A$-module, is injective (cf. also [[Injective module]]). Well-known examples for self-injective algebras are the group algebras
...ence]]) and sends the isomorphism class of a non-projective indecomposable module $Z$ to the starting term $X$.
9 KB (1,240 words) - 08:04, 25 November 2023
...><TR><TD valign="top">[5]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR
A theorem on finiteness of a chain of [[Syzygy|syzygies]]) of a graded module over a ring of polynomials (for the classical formulation see [[#References
18 KB (2,720 words) - 19:17, 19 December 2019
...it. It is a fundamental problem to understand these cohomology groups as a module under this action of the Hecke algebra.
...} ( \Gamma \backslash G ( \mathbf{R} ) \otimes M _ { \mathbf{C} } )$ is a module under the group $G ( \mathbf{R} )$.
12 KB (1,817 words) - 15:30, 1 July 2020
...eory of $ \mathcal D $-modules has found applications in several parts of mathematics, e.g., [[Cohomology|cohomology]] of singular spaces, [[Hodge structure|Hodg
becomes in a natural way a coherent left $ \mathcal D _ {X} $-module. More generally, let $ {\mathcal V} $
24 KB (3,511 words) - 07:03, 10 May 2022
Let $M$ be a [[module]] over a ring and $N$ a submodule.
...top"> Tsit-Yuen Lam, ''Lectures on Modules and Rings'', Graduate Texts in Mathematics '''189''', Springer (1999) {{ISBN|0-387-98428-3}}</TD></TR>
7 KB (1,130 words) - 20:30, 15 November 2023
...ms of linear algebra cease being true if the vector space is replaced by a module. The study of the possibility of generalizations that are also valid for mo
...><TR><TD valign="top">[9]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley
9 KB (1,394 words) - 08:15, 9 January 2024
...e theory of ideals) has found numerous applications in various branches of mathematics.
...for non-associative rings, normal divisors of a group and submodules of a module.
7 KB (1,035 words) - 20:23, 4 April 2020
module $ V ( \Lambda ) $
<TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from Fr
7 KB (992 words) - 09:26, 26 March 2023