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.

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