Simple module: Difference between revisions

Revision as of 17:19, 17 October 2009

Revision as of 17:20, 17 October 2009

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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 ordertheory)|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.