About: Decomposition of a module

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In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple if and only if every module over it is a semisimple module.

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rdfs:label	Decomposition of a module (en)
rdfs:comment	In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple if and only if every module over it is a semisimple module. (en)
dcterms:subject	Module theory
Wikipage page ID	61406091 (xsd:integer)
Wikipage revision ID	1095474196 (xsd:integer)
Link from a Wikipage to another Wikipage	Product ring Pure-injective module Endomorphism ring Module (mathematics) Module theory Permutation Characterization (mathematics) Unit (ring theory) Indecomposable module Artin–Wedderburn theorem Mathematical induction Matrix (mathematics) Quotient module Endomorphism Modular law Converse (logic) Claudio Procesi Ideal (ring theory) Idempotent (ring theory) Krull–Schmidt theorem Matrix ring Central idempotent Local ring Minimal ideal Fitting's lemma Center (ring theory) Direct sum of modules Graduate Texts in Mathematics Kaplansky's theorem on projective modules Length of a module Mathematical proof Ring (mathematics) Isomorphic Countably generated module Abstract algebra Bijection Module homomorphism Division ring Free module Group theory If and only if Integer Semisimple ring Image (mathematics) Semisimple module Simple module Submodule Semisimple decomposition Identity map Uniserial module
Link from a Wikipage to an external page	https://pages.uoregon.edu/anderson/rings/COMPLETENOTES.PDF
sameAs	Decomposition of a module Decomposition of a module Decomposition of a module
dbp:wikiPageUsesTemplate	dbt:Algebra-stub dbt:Citation dbt:Harv dbt:Main dbt:Reflist dbt:Short_description dbt:Isbn
has abstract	In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple if and only if every module over it is a semisimple module. An indecomposable module is a module that is not a direct sum of two nonzero submodules. Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism rings, then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in group theory, is known as the Krull–Schmidt theorem. A special case of a decomposition of a module is a decomposition of a ring: for example, a ring is semisimple if and only if it is a direct sum (in fact a product) of matrix rings over division rings (this observation is known as the Artin–Wedderburn theorem). (en)
prov:wasDerivedFrom	wikipedia-en:Decomposition_of_a_module?oldid=1095474196&ns=0
page length (characters) of wiki page	14565 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Decomposition_of_a_module
is rdfs:seeAlso of	Semisimple representation
is Link from a Wikipage to another Wikipage of	Azumaya's theorem Glossary of module theory Direct sum of modules Indecomposable decomposition
is Wikipage redirect of	Azumaya's theorem Indecomposable decomposition
is foaf:primaryTopic of	wikipedia-en:Decomposition_of_a_module

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