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
...alpha $ ; furthermore, the sum of all $ G $ -submodules in the $ G $ -module $ L _{2} (G) $ that are isomorphic to $ R ^ \alpha $ is precisely th
...><TD valign="top">[a3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from Fr
8 KB (1,117 words) - 20:03, 27 February 2021
...ly-generated $A$-module with $\dim_AM = s$. Then $M$ is called a Buchsbaum module if the difference
...ively, $e _ { \mathfrak{q} } ^ { 0 } ( M )$) denotes the length of the $A$-module $M / \mathfrak { q } M$ (respectively, the multiplicity of $M$ with respect
27 KB (4,003 words) - 17:43, 1 July 2020
module $ \Omega ^ {1} ( A ) $
module, is equal to the cohomology of the complex of alternating $ A $-
8 KB (1,172 words) - 19:18, 17 March 2023
...$-modules and $A$-bimodules are equivalent via the functors sending an $R$-module $N$ to $A\otimes N$, and sending an $A$-bimodule $M$ to
...uren, "Théorie de la descente et algèbres d'Azumaya", ''Lecture Notes in Mathematics'', '''389''', Springer (1974) {{MR|0417149}} {{ZBL|0284.13002}}
4 KB (552 words) - 16:03, 26 April 2012
module. $ {\mathcal M} _ {*} $
...Forms in Algebraic Topology (Proc., Princeton 1986)'' , ''Lecture Notes in Mathematics'' , '''1326''' , Springer (1988) pp. 55–68 {{MR|0970281}} {{ZBL|0649.5702
4 KB (561 words) - 19:37, 5 June 2020
The vector spaces most often employed in mathematics and in its applications are those over the field $C$ of [[complex number]]s
...[[ring]] — a vector space is a [[unitary module]] over a field. A unitary module over a non-commutative [[skew-field]] is also called a vector space over a
14 KB (2,558 words) - 11:28, 21 June 2016
...eory. In a wide sense, the term "K-theory" is used to denote the branch of mathematics that includes algebraic $ K $-theory and topological $ K $-theory, and
...ntinuous functions, which turns out to be a [[Projective module|projective module]].
17 KB (2,459 words) - 07:32, 26 February 2022
...to them (ideals, modules, valuations, etc., cf. [[Ideal|Ideal]]; [[Module|Module]]; [[Valuation|Valuation]]).
...ective module]]; [[Injective module|Injective module]]; [[Flat module|Flat module]]). The study of an arbitrary class of rings is closely related to the stud
16 KB (2,400 words) - 17:45, 4 June 2020
...the term refers more specifically to the closely related operators used in mathematics as a powerful tool in many applications, notably, constructing certain repr
Vertex operators arose in mathematics in the following construction of the "basic" highest weight representatio
8 KB (1,142 words) - 16:46, 1 July 2020
...uctures do not coincide, one has to distinguish between left and right $A$-module morphisms. After settling for a convention, say of considering the (symmetr
...ing the dual module, one must distinguish again between left and right $A$-module morphisms into $A$. Having fixed one choice of dual, say dual on the left,
18 KB (2,722 words) - 17:47, 1 July 2020
...} ^ { 2 } ( S _ { j } , S _ { i } )$, where $S _ { t }$ is the simple $R$-module associated to the vertex $t \in Q_0$. Then the definition of $q_{ R}$ depen
18 KB (2,636 words) - 06:50, 15 February 2024
A tensor is similarly defined on an arbitrary unitary module $ V $
is a free or a finitely-generated free module.
13 KB (1,934 words) - 08:25, 6 June 2020
module $ M $,
...R><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
5 KB (820 words) - 19:36, 5 June 2020
...or is a non-degenerate reflexive form; sometimes $E$ is taken to be a free module of finite type. Often one means by classical groups other groups closely re
In the more general situation when $E$ is a module over a ring $K$ the results on classical groups are not so exhaustive (see
12 KB (1,991 words) - 18:05, 29 November 2014
.../TD> <TD valign="top"> P.S. Urysohn, "Works on topology and other areas of mathematics" , '''1–2''' , Moscow-Leningrad (1951) (In Russian)</TD></TR></table>
A number associated to a ring or module in such a way that its behaviour under some classical operations, e.g. subo
38 KB (5,928 words) - 19:35, 5 June 2020
where $\textbf{Z}/p[G/G_i]$ denotes the usual permutation module obtained by induction. When concatenated together, one obtains a representa
...valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , ''Lecture Notes in Mathematics'' , '''5''' , Springer (1994) (Edition: Fifth) {{MR|1324577}} {{ZBL|0812.12
6 KB (868 words) - 22:16, 5 February 2021
A typical example is a basis of a (unitary) module $ M $
module $ M $
28 KB (4,564 words) - 07:37, 26 March 2023
module. The subalgebra $ B $
module do not depend on the choice of $ B $.
16 KB (2,336 words) - 08:02, 6 June 2020
The branch of mathematics in which one studies algebraic operations (cf. [[Algebraic operation|Algebr
...algebra, its principal branches and its connection with other branches of mathematics.==
17 KB (2,478 words) - 16:09, 1 April 2020
Now, let $R$ be the $\bf Z$-module of all formal sums $\sum c _ { \alpha } D \alpha D$, i.e. the free Abelian
Modular forms turn up all over mathematics and physics and, hence, so do the Hecke operators. See the references for a
6 KB (1,050 words) - 00:38, 15 February 2024
...thematical structures of the same type. Correspondences are widely used in mathematics and also in various applied disciplines, such as theoretical programming, g
Thus one has group correspondences, module correspondences, ring correspondences, and others. Such correspondences oft
8 KB (1,265 words) - 17:31, 5 June 2020
module, then $ V = \oplus_{i=1}^ {n} V _ {i} $,
...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from Fr
10 KB (1,457 words) - 18:48, 13 January 2024
...and $H$-module-$K$ to mean respectively left $\Ql H$-module, right $\Ql H$-module and $(\Ql H,\Ql K)$-bimodule.
...ume $M$ is an $\mbf{L}^F$-module then we define a virtual left $\mbf{G}^F$-module
48 KB (8,458 words) - 18:22, 13 August 2023
...up of automorphisms of a skew-symmetric bilinear form $\Phi$ on a left $K$-module $E$, where $K$ is a commutative ring (cf.
...ign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Algebra", ''Elements of mathematics'', '''1''', Addison-Wesley (1973) (Translated from French) {{MR|0354207}}
6 KB (1,078 words) - 14:22, 3 November 2013
...also [[Modular lattice|Modular lattice]]). A complete lattice is called a module over a multiplicative lattice $ L $
...R><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
8 KB (1,218 words) - 08:02, 6 June 2020
mathematics, especially so with algebraic geometry and algebraic
module. All semi-simple associative rings are a subdirect sum of
11 KB (1,726 words) - 20:09, 15 December 2020
..." /> (cf. also [[Homotopy group|Homotopy group]]; [[Crossed module|Crossed module]]; [[Crossed complex|Crossed complex]]; [[Fundamental group|Fundamental gro
24 KB (3,338 words) - 17:29, 7 February 2011
module, where $ \mathfrak g $
module $ \phi $
32 KB (4,602 words) - 04:46, 7 January 2022
...nough information. The most important algebraic invariant is the Alexander module, defined as follows. The [[Homology group|homology group]] $ H _ {1} ( M
has the natural structure of a module over $ \Lambda _ \mu = \mathbf Z [ \mathbf Z ^ \mu ] $(
37 KB (5,595 words) - 17:15, 18 May 2024
...V(\mathcal{E}) $ corresponds bijectively to the set of $ \mathcal{O}_{X} $-module homomorphisms $ \mathcal{E} \to {f^{*}}(\mathcal{O}_{Y}) $, where $ f $ is
...rname{Spec}(A) $, then algebraic vector bundles correspond to [[Projective module|projective modules]] of finite type over the ring $ A $. If the rank of the
14 KB (2,169 words) - 08:12, 14 December 2016
...$ in 1981, intriguing connections between $\mathcal{M}$ and other areas of mathematics had been noted:
...$V ^ { \natural } = \oplus _ { n } V _ { n }$, now known as the moonshine module, such that
21 KB (3,107 words) - 09:12, 10 November 2023
...ely reducible (in other words: every finite-dimensional $ \mathfrak g $ -module is semi-simple);
...rak g $ with values in an arbitrary finite-dimensional $ \mathfrak g $ -module is trivial.
25 KB (3,293 words) - 19:24, 26 March 2023
..._\mathcal{A}G_\QQ \backslash G_\mathcal{A})$ such that the $G_\mathcal{A}$-module $\pi_f=\otimes_p \pi_p$ generated by the right $G_\mathcal{A}$-translates o
...lign="top"> D. Hejhal, "The Selberg trace formula II" , ''Lecture Notes in Mathematics'' , '''1001''' , Springer (1983) {{MR|}} {{ZBL|0543.10020}} </TD></TR>
11 KB (1,656 words) - 06:13, 1 April 2023
...polynomial by choosing an element of the Jones–Conway [[Skein module|skein module]] of the solid torus (a linear combination of links in a solid torus) and c
...mials" , ''Ideas from Graph Theory and Statistical Mechanics, Panoramas of Mathematics'' , '''34''' , Banach Center Publ. (1995) pp. 121–148</td></tr>
18 KB (2,713 words) - 05:14, 15 February 2024
module $ M $(
cf. also [[Module|Module]]) and any lattice of subobjects of an object in an [[Abelian category|Abel
29 KB (4,201 words) - 16:31, 9 December 2023
form a locally free module $ \Gamma ( \pi ) $
...etry; by now it has become a basic tool for studies in various branches of mathematics: differential and algebraic topology, the theory of linear connections, alg
11 KB (1,695 words) - 20:37, 23 December 2023
module $ A $.
...R><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</T
13 KB (2,163 words) - 22:11, 5 June 2020
...igns to each open subset $U\subseteq X$ an Abelian group $F(U)$ (a ring, a module over a ring, etc.) and to every pair of open sets $V\subseteq U$ a homomorp
$\G_c(X,\cF)$) be the group (ring, module, etc.) of all sections of
26 KB (4,342 words) - 15:06, 15 July 2014
where $(v)$ is a fixed basis of a free $A$-module $E$ of finite rank over the commutative ring $A$ (with a unit element). If
[[Fractional ideal|fractional ideal]] in $B$. Then the $A$-module $D(\fb)$ generated by all discriminants of the form $D(w_1,\dots,w_n)$, whe
16 KB (2,947 words) - 08:53, 9 December 2016
a unitary $k$-module $L$ over a commutative ring $k$ with a unit that is endowed with a bilinear
Lie algebras appeared in mathematics at the end of the 19th century in connection with the study of Lie groups (
25 KB (4,037 words) - 07:06, 23 April 2016
...structures are often encountered in functional analysis and other areas of mathematics. For further details on rings with additional structures see [[Normed ring|
...><TR><TD valign="top">[4]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR
21 KB (3,225 words) - 09:25, 13 July 2022
...lgebras and their representations are deeply related to many directions in mathematics and physics, in particular, the representation theory of the Fischer–Grie
...rator realization of the Griess algebra, of what they called a "moonshine module" $V ^ { \natural }$ for $\bf M$ having the desired relation with $J ( q )$
20 KB (2,919 words) - 00:57, 15 February 2024
...om that on the generic fiber, e.g., sections of the vector bundle form a [[module]] over the ring of "scalar" functions.
...th notes by S. Ramanan. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, '''20''', 1965. Reprinted by Springer-Verlag, Berlin, 1986.
32 KB (5,299 words) - 12:27, 12 December 2020
...dean geometry is not the only possible one, and that it is advantageous in mathematics to develop other, non-Euclidean, geometries, irrespective of their relation
...ring]] and [[Differential operator on a module|Differential operator on a module]]). Such an approach makes it possible to generalize various results of dif
30 KB (4,323 words) - 19:35, 5 June 2020
The area of mathematics whose main object of study is the index of operators (cf. also [[Index of a
...ge number of applications of index theorems to topology and other areas of mathematics. A few examples follow. In [[#References|[a12]]], Atiyah and W. Schmid used
35 KB (5,243 words) - 17:45, 1 July 2020
may also be a module, and if the manifold is not orientable, the theorem is true modulo 2. Repla
<TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French
64 KB (9,418 words) - 12:44, 8 February 2020