Number of submodules of a semisimple module

Question

Let $M$ be a semisimple $A$-module, where $A$ is an associative algebra. I want to find the number of submodules of $M$.

First suppose all the composition factors of $M$ are pairwise non-isomorphic, say $M = S_1 \oplus S_2 \oplus \cdots \oplus S_k$ with $S_i$ simple and $S_i \not\cong S_j$ if $i \neq j.$ We know that any submodule of $M$ has to be a direct summand of $M$, so there are $2^k$ different submodules.

What happens if $S_i \cong S_j$ for some $i \neq j?$ In particular, suppose $M = S \oplus S.$ Would the situation be any different from above? The only submodules I can think of are $\{0\},S,M$, but surely there are semisimple algebras with infinitely many submodules?

Answer 1

I know that you did this exam today so might not care, but just in case you do, as in question 4 on your second representation theory sheet, if we have an isomorphism $\phi : S_i \to S_j$ then we have a homomorphism given by $x \mapsto x + \lambda \phi(x)$. Where $\lambda$ is an element in our field.
All these images are isomorphic to $S_i$ but distinct for different $\lambda$ in the field.

Answer 2

Take into account that a vector space on a not finite field is semisimple and has infinite linear subspaces.