Simple module

A (left) $R$-module $M\ne 0$ is simple if its only submodules are $0$ and $M$. Equivalently, $M$ has no proper nonzero submodule.

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 ).

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 $2\mathbb Z/4\mathbb Z$.