Simple module: Difference between revisions

Revision as of 19:10, 14 March 2020

Revision as of 19:40, 1 June 2020

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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|''f'' : ''M'' → ''N''}} be a module homomorphism. If ''M'' is simple, then ''f'' is either the zero homomorphism or [[injective]] because the 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 {{nowrap1=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'''.