Editing Simple module (section)

In [[mathematics]], specifically in [[ring theory]], the '''simple modules''' over a [[ring (mathematics)|ring]] ''R'' are the (left or right) [[module (mathematics)|module]]s over ''R'' that are [[zero module|non-zero]] and have no non-zero proper [[submodule]]s.  Equivalently, a module ''M'' is simple [[if and only if]] every [[cyclic module|cyclic submodule]] generated by a {{nowrap|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''.