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