Semisimple module

A module $M$ is semisimple if it is (isomorphic to) a direct sum of simple modules . Equivalently, every short exact sequence $0\to A\to M\to B\to 0$ is split . Another standard characterization is that every submodule of $M$ is a direct summand; see semisimple implies direct summand .

Semisimple modules are precisely those with completely reducible submodule structure; they are the module-theoretic analog of diagonalizable operators in linear algebra.

Examples:

• Any vector space over a field is semisimple (as a module over that field).
• As a $\mathbb Z$-module, $(\mathbb Z/p\mathbb Z)^n$ is semisimple for any prime $p$ and integer $n\ge 1$.
• (Nonexample) $\mathbb Z/p^2\mathbb Z$ is not semisimple as a $\mathbb Z$-module.