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.