Adjective
(mathematics, of a module) In which each submodule is a direct summand.
(mathematics, of an algebra or ring) diagonalizable.
(mathematics, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.
(mathematics, of a Lie algebra) Being a direct sum of simple Lie algebras.
(mathematics, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial.
Source: en.wiktionary.orgA Lie algebra is solvable if and only if Classification The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. Source: Internet
A Lie algebra is called semisimple if its radical is zero. Source: Internet
Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras. Source: Internet
The problem is to prove that the 2-layer of the centralizer of an involution in a simple group is semisimple. Source: Internet
In particular, a simple Lie algebra is semisimple. Source: Internet
Semisimple rings A ring is called a semisimple ring if it is semisimple as a left module (or right module) over itself; i.e., a direct sum of simple modules. Source: Internet