Simple module - Imported Knowlpedia Content

A (left) $R$ -module $M\ne 0$ is simple if its only submodules are $0$ and $M$ .

Simple modules are the analogs of "atoms" in module theory: they appear as factors in composition series, and semisimplicity is built from direct sums of simples. Over a commutative ring, simple modules are precisely $R/\mathfrak m$ where $\mathfrak m$ is a maximal ideal, hence are vector spaces over the field $R/\mathfrak m$ (compare quotients).

Equivalent characterizations

Equivalently, $M$ has no proper nonzero submodule.

Examples

As a $\mathbb Z$ -module, $\mathbb Z/p\mathbb Z$ is simple for prime $p$ .
If $k$ is a field, then any 1-dimensional $k$ -vector space is a simple $k$ -module.
(Nonexample) $\mathbb Z/4\mathbb Z$ is not simple as a $\mathbb Z$ -module because it has the nontrivial submodule $2\mathbb Z/4\mathbb Z$ .