Simple module: Difference between revisions

Revision as of 21:59, 7 July 2013

Revision as of 19:02, 3 March 2014

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In [[mathematics]], specifically in [[ring theory]], the '''simple modules''' over a ring ''R'' are the (left or right) [[module (mathematics)|module]]s over ''R'' whichthat have no non-zero proper submodules. Equivalently, a module ''M'' is simple [[if and only if]] every [[cyclic module|cyclic submodule]] generated by a non-zero element of ''M'' equals ''M''. Simple modules form building blocks for the modules of finite [[length of a module|length]], and they are analogous to the [[simple group]]s in group theory.

In this article, all modules will be assumed to be right [[unital module]]s over a ring ''R''.