Semisimple module

Remarks

The empty family is allowed, so the zero module is semisimple. Equivalently, $M$ is a sum of simple submodules, or every submodule of $M$ is a direct summand. Example: the $\Z$-module $\Z/6\Z$ is semisimple. The check: $\Z/6\Z = \set{0, 3} \oplus \set{0, 2, 4}$, and these subgroups have prime orders $2$ and $3$, so they are simple. Non-example: the $\Z$-module $\Z/4\Z$ is not semisimple. Its only simple submodule is $\set{0, 2}$, so every sum of simple submodules lies inside $\set{0, 2} \ne \Z/4\Z$. Every vector space over a field, or over a division ring, is semisimple, as the direct sum of the lines spanned by the vectors of a basis.