In mathematics, the term semisimple (sometimes completely reducible) is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' ('irreducible') parts, that fit together in the cleanest way (by direct sum). Briefly, "semisimple = direct sum of simple", or equivalently "completely reducible = direct sum of irreducible".
- A semisimple module is one in which each submodule is a direct summand.
- A semisimple algebra (or ring) is one that is semisimple as a module over itself.
- A semisimple operator (or matrix) is one for which every invariant subspace has an invariant complement. This is equivalent to the minimal polynomial being square-free. Over an algebraically closed field it is equivalent to diagonalizable.
- A semisimple Lie algebra is a Lie algebra that is a direct sum of simple Lie algebras.
- A semisimple algebraic group is a linear algebraic group whose radical of the identity component is trivial.