## Semisimple Modules Equivalent Conditions

Proposition:  For a right $A$-module $M$, the following are equivalent:

(i) $M$ is semisimple.

(ii) $M=\sum\{N\in S(M): N\ \text{is simple}\}$.

(iii) $S(M)$ is a complemented lattice, that is, every submodule of $M$ has a complement in $S(M)$.

We are following Pierce’s book’s Associative Algebras (Graduate Texts in Mathematics) notation, where $S(M)$ is the set of all submodules of $M$.

This proposition is can be used to prove two useful Corollaries:

Corollary 1) If $M$ is semisimple and $P$ is a submodule of $M$, then both $P$ and $M/P$ are semisimple. In words, this means that submodules and quotients of a semisimple module is again semisimple.

Proof: Since $M$ is semisimple, by (iii) we have $M\cong P\oplus P'$, where $P'\cong M/P$. Let $N$ be a submodule of $P\leq M$. Then $N$ has a complement $N'$ in $S(M)$: $M=N\oplus N'$. $P=P\cap(N\oplus N')=N\oplus (N'\cap P)$ with $N'\cap P\in S(P)$. Thus, we have that $P$ is semisimple by condition (iii). Similarly, $M/P\cong P'$ 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. 