Proposition: For a right -module , the following are equivalent:
(i) is semisimple.
(iii) is a complemented lattice, that is, every submodule of has a complement in .
We are following Pierce’s book’s Associative Algebras (Graduate Texts in Mathematics) notation, where is the set of all submodules of .
This proposition is can be used to prove two useful Corollaries:
Corollary 1) If is semisimple and is a submodule of , then both and are semisimple. In words, this means that submodules and quotients of a semisimple module is again semisimple.
Proof: Since is semisimple, by (iii) we have , where . Let be a submodule of . Then has a complement in : .
with . Thus, we have that is semisimple by condition (iii). Similarly, is semisimple.
Corollary 2) A direct sum of semisimple modules is semisimple.
Proof: This is quite clear from the definition of semisimple modules being direct sum of simple modules. A direct sum of (direct sum of simple modules) is again a direct sum of simple modules.