Simple module

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

In this article, all modules will be assumed to be right unital modules over a ring R.

Examples

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 groups of prime 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.