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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