semisimple

Adjective

semisimple (not comparable)

1. (most generally) Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects.
2. (module theory) In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules.
3. (ring theory) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.
4. (proscribed, ring theory) Semiprimitive: having trivial Jacobson radical.

   Synonyms:

   Jacobson semisimple

5. For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.
6. (Lie theory) Being a direct sum of simple Lie algebras.
7. (representation theory) Being a direct sum of simple representations (also known as irreducible representations).
8. Being a linear algebraic group whose radical of the identity component is trivial.