Search results
Jump to navigation
Jump to search
Page title matches
- [[Module theory]] is the branch of mathematics in which [[module (mathematics)|modules]] are studied. This is a glossary of some terms of th ...gebra]]'', ''[[Glossary of ring theory]]'', ''[[Glossary of representation theory]]''. ...21 KB (3,042 words) - 00:51, 10 February 2025
Page text matches
- ...ule''' is a [[module (mathematics)|module]] which is close to being [[free module|free]]. ...ted module]] ''F'' over ''R'' such that <math> M \oplus F</math> is a free module. ...976 bytes (144 words) - 12:11, 18 June 2024
- ...ge Lang|date=2002|publisher=Springer|isbn=978-0387953854}}</ref> The dual module is typically denoted ''M''<sup>∗</sup> or {{nowrap|Hom<sub>''R''</sub>(''M' If the base ring ''R'' is a [[field (mathematics)|field]], then a dual module is a [[dual vector space]]. ...1 KB (207 words) - 16:16, 2 February 2024
- ...'k'', the '''module of covariants''' of type ''M'' is the <math>A^G</math>-module [[Category:Module theory]] ...815 bytes (113 words) - 19:50, 12 August 2023
- ...ebra]], an '''invertible module''' is intuitively a [[module (mathematics)|module]] that has an [[inverse element|inverse]] with respect to the [[tensor prod ...rime element|primes]] ''P'' of ''R''. Now, if ''M'' is an invertible ''R''-module, then its [[Duality (mathematics)#Dual objects|dual]] {{nowrap|''M''<sup>*< ...1 KB (184 words) - 04:53, 4 May 2024
- ...|finite]] generating set, then a module is said to be [[finitely generated module|finitely generated]]. This applies to [[ideal (ring theory)|ideals]], which are the submodules of the ring itself. In particular, a [[ ...3 KB (534 words) - 21:43, 12 August 2023
- ...ntation is a <math>(\mathfrak{g},K)</math>-module, then its Harish-Chandra module is a representation with desirable factorization properties. ...ath>(\pi,V)</math> is a representation of ''G'', then the ''Harish-Chandra module'' of <math>\pi</math> is the subspace ''X'' of ''V'' consisting of the [[K- ...2 KB (361 words) - 20:08, 22 March 2021
- ...irst=Joachim|date=1964|title=A Module is Flat if and Only if its Character Module is Injective|journal=[[Canadian Mathematical Bulletin]]|language=en|volume= ...e=189|location=New York, NY}}</ref><ref name=":3">{{Cite book|title=Module theory, extending modules and generalizations|last1=Tercan|first1=Adnan|last2=Yüce ...6 KB (989 words) - 17:22, 18 February 2025
- ...vention. In other words, it is a [[chain complex]] having a structure of a module, while a [[differential graded algebra]] is a chain complex with a structur ...alent to that of a '''simplicial module'''; "equivalent" in the [[category theory|categorical]] sense; see {{sectionlink||The Dold–Kan correspondence}} below ...2 KB (326 words) - 03:21, 13 May 2024
- {{distinguish|Torsionless module}} ...or|regular element]] (non [[zero-divisor]]) of the ring. In other words, a module is ''torsion free'' if its [[torsion submodule]] contains only the zero ele ...4 KB (672 words) - 14:20, 10 November 2024
- ...X'' equipped with an ''S''-action is exactly a [[nlab:module+over+a+monoid|module over a monoid]] in ''Cat''. ...s.edu/~weibel/Kbook.html |title=The K-book: an introduction to algebraic K-theory |year=2013 |publisher=American Mathematical Society |series=Graduate Studie ...1 KB (217 words) - 16:31, 18 February 2025
- ...the field of [[ring theory]], a '''lattice''' is a [[module (mathematics)|module]] over a [[ring (mathematics)|ring]] that is [[embedding|embedded]] in a [[ ...''K''-vector space ''V'' is a ''lattice'' if ''M'' is [[finitely generated module|finitely generated]] over ''R''. It is ''full'' if {{nowrap|1=''V'' = ''K' ...2 KB (233 words) - 15:36, 25 September 2023
- ...a)|resolution]] of the module by [[flat module]]s. The weak dimension of a module is, at most, equal to its [[projective dimension]]. ...^R(M,N)</math> is nonzero for some right ''R''-module ''M'' and left ''R''-module ''N''. If there is no such largest number ''n'', the weak global dimension ...3 KB (492 words) - 23:54, 12 August 2023
- {{distinguish|Torsion-free module}} In [[abstract algebra]], a [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|ring]] ''R'' is called '''torsionless'' ...5 KB (737 words) - 11:48, 9 February 2024
- ..._a_ring|quotient]] of the [[tensor algebra]] of ''M'' by the [[ideal (ring theory)|ideal]] ''I'' generated by elements of the form: ...ic algebra of ''V''. Similarly, the graded-symmetric algebra of the graded module with ''V'' in degree one and zero elsewhere is the exterior algebra of ''V' ...2 KB (242 words) - 03:40, 13 May 2024
- In [[representation theory]], the '''stable module category''' is a [[category (mathematics)|category]] in which projectives a ...]. The stable module category is defined by setting the [[object (category theory)|objects]] to be the ''R''-modules, and the [[morphism]]s are the [[equival ...4 KB (653 words) - 17:33, 17 November 2020
- ...h>G \to \operatorname{Aut}(X)</math>; cf. [[Automorphism group#In category theory]].</ref> of the [[symmetric group]] <math>\mathbb{S}_n</math>.<!-- The noti == S-module == ...2 KB (339 words) - 18:32, 31 July 2024
- ...ath>''') is a [[category (mathematics)|category]] whose [[Object (category theory)|objects]] are certain [[Lie algebra representation|representations]] of a If <math>M</math> is a <math>\mathfrak{g}</math>-module and <math>\lambda\in\mathfrak{h}^*</math>, then <math>M_\lambda</math> is t ...3 KB (517 words) - 22:31, 24 February 2021
- ..., the three module conditions [[noetherian module|Noetherian]], [[artinian module|Artinian]] and "has a [[composition series]]" are equivalent. Without the ...is not true in general for Artinian modules, because there are [[Artinian module#Relation to the Noetherian condition|examples of Artinian modules which are ...5 KB (671 words) - 13:33, 11 October 2024
- ...''M'' is a non-zero [[finitely generated module|finitely generated]] ''R''-module of finite [[projective dimension]], then: ...imension of a module, and depth for the [[depth (ring theory)|depth]] of a module. ...2 KB (264 words) - 23:38, 12 August 2023
- ...y''' concerns the '''multiplicity of a module''' ''M'' at an [[Ideal (ring theory)|ideal]] ''I'' (often a maximal ideal) ...la, it is linked to an [[intersection multiplicity]] in the [[intersection theory]]. ...3 KB (417 words) - 13:44, 9 August 2023