Simple module: Difference between revisions

Revision as of 10:53, 15 August 2009

Revision as of 22:27, 16 August 2009

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In this article, all modules are assumed right [[Unital module|unital]] modules over some ring ''R'' (as opposed to "left module over ''R''").

== Examples ==

[[Abelian group]]s are the same as [[integer|'''Z''']]-modules. The simple '''Z'''-modules are precisely the [[cyclic group]]s of [[prime number|prime]] [[group order|order]].

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Given a ring ''R'' and a [[left ideal]] ''I'' in ''R'' then ''I'' is a simple ''R''-module if and only if ''I'' is a minimal left ideal in ''R'' (does not contain any other non trivial left ideals). The [[factor module]] ''R''/''I'' is a simple ''R''-module [[if and only if]] ''I'' is a maximal left ideal in ''R'' (is not contained in any other non-trivial left ideals).

== Properties ==

The simple modules are precisely the modules of [[length of a module|length]] 1; this is a reformulation of the definition.

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Not every module has a simple submodule; consider for instance the '''Z'''-module '''Z''' in light of the first example above.

Let ''M'' and ''N'' be (left or right) modules over the same ring, and let {{nowrap begin}}''f'' : ''M'' →→ ''N''{{nowrap end}} be a [[module homomorphism]]. If ''M'' is simple, then ''f'' is either the zero homomorphism or [[injective]] because the [[kernel (algebra)|kernel]] of ''f'' is a submodule of ''M''. If ''N'' is simple, then ''f'' is either the zero homomorphism or [[surjective]] because the [[image (mathematics)|image]] of ''f'' is a submodule of ''N''. If {{nowrap begin}}''M'' = ''N''{{nowrap end}}, then ''f'' is an [[endomorphism]] of ''M'', and if ''M'' is simple, then the prior two statements imply that ''f'' is either the zero homomorphism or an isomorphism. Consequently the [[endomorphism ring]] of any simple module is a [[division ring]]. This result is known as '''[[Schur's lemma]]'''.

The converse of Schur's lemma is not true in general. For example, the '''Z'''-module [[rational number|'''Q''']] is not simple, but its endomorphism ring is isomorphic to the field '''Q'''.

== See also ==

* [[Semisimple module]]s are modules that can be written as a sum of simple submodules

* [[Simple group]]s are similarly defined to simple modules

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==Notes==

==References==

{{Reflist}}

==See also==

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*[[Artin-Wedderburn theorem]]

{{DEFAULTSORT:Simple Module}}

[[Category:moduleModule theory]]

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