''left topological module''
...lian topological group can be given the natural structure of a topological module over the completion $\hat R$ of the ring $R$.
1 KB (204 words) - 18:09, 17 April 2017
...clopediaofmath.org/legacyimages/c/c025/c025920/c0259201.png" /> of a right
module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...//www.encyclopediaofmath.org/legacyimages/c/c025/c025920/c02592016.png" />-
module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
6 KB (880 words) - 17:29, 7 February 2011
...ry decomposition. Any module of finite type over a Lasker ring is a Lasker module. E. Lasker [[#References|[1]]] proved that there is a primary decomposition
...R><TD valign="top">[3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
1 KB (157 words) - 11:43, 29 June 2014
...the discipline is the notion of skein module, i.e., the quotient of a free module over ambient isotopy classes of links in a manifold by properly chosen loca
563 bytes (84 words) - 09:12, 4 May 2014
...l prime ideals $(p)$, where $p$ divides the order of $M$. For an arbitrary module $M$ the set $\mathrm{Supp}(M)$ is empty if and only if $M=0$.
<TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
918 bytes (149 words) - 20:46, 18 October 2014
[[Module|module]] $M$ into a (left) module $N$ over the same ring $A$, satisfying the conditions
...$ is also known as an anti-linear mapping. A semi-linear mapping of an $A$-module $M$ into itself is known as a semi-linear transformation.
2 KB (264 words) - 22:23, 2 November 2013
...the ring $A$ such that $\sum_{i\in I} a_i m_i = 0$. Thus there arises the module of syzygies, the chain complex of syzygies, etc. See also [[Hilbert syzygy
...gular algebras and regular sequences, cf. [[Koszul complex]]; [[Depth of a module]].
906 bytes (150 words) - 16:48, 23 November 2023
$#C+1 = 38 : ~/encyclopedia/old_files/data/A010/A.0100860 Adjoint module,
''contragradient module, dual module''
3 KB (435 words) - 16:09, 1 April 2020
...free module by a submodule which is an image of a finitely-generated free module. Such a ring is called a left coherent ring; a right coherent ring is defin
module is finitely presentable; 2) a direct product of flat right $ R $-
3 KB (479 words) - 17:45, 4 June 2020
...e topology is said to be left linear). Similarly, a topology on a left $A$-module $E$ is linear if there is a fundamental system of neighbourhoods of zero co
...E'$, where $x\in E$ and $E'$ is a submodule of $E$) has a limit point. Any module of finite type over a complete local Noetherian ring is linearly compact.
2 KB (315 words) - 19:21, 14 August 2014
$#C+1 = 28 : ~/encyclopedia/old_files/data/F040/F.0400150 Filtered module
A module $ M $
3 KB (480 words) - 16:44, 13 January 2024
...s the conductor is regarded as an ideal of $\bar A$. If $\bar A$ is an $A$-module of finite type (e.g., if $A$ is a [[geometric ring]]), a [[prime ideal]] $\
<TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
1 KB (161 words) - 08:20, 30 November 2014
$#C+1 = 18 : ~/encyclopedia/old_files/data/I052/I.0502490 Invertible module
A [[module]] $M$ over a [[commutative ring]] $A$
2 KB (308 words) - 18:37, 11 April 2023
...ciative ring $A$ with unit is the algebra $T(V)$ over $A$ whose underlying module has the form
If the module $V$ is free and finitely generated, then $T(V^*)$ is naturally isomorphic t
2 KB (329 words) - 04:49, 23 July 2018
module $ M $
module.
3 KB (559 words) - 16:09, 1 April 2020
''of a module''
...n be represented as a direct sum of simple modules. [[Completely-reducible module]]s (semi-simple modules) can be characterized as modules that coincide with
2 KB (371 words) - 15:04, 19 November 2023
module $ E $''
module $ L _ {n} ( E, A) $,
6 KB (855 words) - 20:46, 16 January 2024
...e $E_1$ of the [[injective hull]] $E$ of the module $M/N$ (cf. [[Injective module]]) the intersection of the kernels of the homomorphisms from $E$ into $E_1$
<TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
3 KB (455 words) - 19:51, 5 October 2017
''of a left module $M$ over a ring $R$''
...$. In contrast to the dual notion of an injective envelope (an [[injective module]] $Q$ together with an essential monomorphism $M \rightarrow Q$) projective
2 KB (368 words) - 16:20, 18 November 2023
''of a module $M$''
...Anderson, K.R. Fuller, "Rings and Categories of Modules" Graduate Texts in Mathematics '''13''' Springer (2012) {{ISBN|1468499130}}
480 bytes (76 words) - 14:23, 12 November 2023
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