**Proposition: ** For a right -module , the following are equivalent:

(i) is semisimple.

(ii) .

(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.

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