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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