Direct sum of isomorphic simple modules

Question

Let $R$ be a ring and $M$ an $R$-module.
Suppose that $\{M_i\}_{i\in I}$ is a (possibly infinite) collection of simple submodules of $M$, which are pairwise isomorphic.
Suppose that $M$ is the direct sum of $\{M_i\}_{i\in I}$.
Suppose further that $M$ is also the direct sum of $\{K_j\}_{j\in J}$, where each $K_j$ is a simple submodule of $M$.

I can prove that each $K_j$ is isomorphic to each $M_i$. However, is it true that $I$ and $J$ are of the same cardinality? If so, I'd appreciate a direct proof (using basic equivalent definitions of a semisimple module is fine).

Answer 1

This is a special case of the general Krull-Schmidt-Remak-Azumaya theorem.

Theorem 2.12 (Krull-Schmidt-Remak-Azumaya Theorem) Let $M$ be a module that is a direct sum of modules with local endomorphism rings. Then any two direct sum decompositions of $M$ into indecomposable direct summands are isomorphic.

There is an elementary proof of the special case based on a generalization of the concept of dimension of vector spaces, which you can find, for instance, in Jacobson's “Basic Algebra II”.