Simple module: Difference between revisions

Revision as of 17:12, 18 March 2013

Revision as of 13:28, 10 April 2013

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== Examples ==

'''[[Integer|'''Z]]''']]-modules are the same as abelian groups, so a simple '''Z'''-module is an abelian group which has no non-zero proper subgroups. These are the [[cyclic group]]s of [[prime number|prime]] [[order (group theory)|order]].

If ''I'' is a right ideal of ''R'', then ''I'' is simple as a right module if and only if ''I'' is a minimal non-zero right ideal: If ''M'' is a non-zero proper submodule of ''I'', then it is also a right ideal, so ''I'' is not minimal. Conversely, if ''I'' is not minimal, then there is a non-zero right ideal ''J'' properly contained in ''I''. ''J'' is a right submodule of ''I'', so ''I'' is not simple.

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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'''.

== Simple modules and composition series ==