Definition of "semisimple"

(mathematics, of a module) In which each submodule is a direct summand.  examples

(mathematics, of an algebra or ring) diagonalizable.  examples

(mathematics, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.  examples

(mathematics, of a Lie algebra) Being a direct sum of simple Lie algebras.  examples

(mathematics, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial.  examples