Semisimple modules

Question

Let $M$ be a semisimple $R$-module, where $R$ is a ring. By definition, $M$ is isomorphic to $\oplus_{i=1}^n S_i$, where $S_i$ are simple $R$-modules. Why is the direct sum finite?

Answer 1

It frequently isn't finite.

For example $\oplus_{i=1}^\infty F$ is a semisimple module that isn't a finite direct sum.

The case for $R_R$ itself being semisimple is the subject of this duplicate, which I think you are referring to in the comments. Certainly the same cannot be said for arbitrary modules.