# Direct sum of isomorphic simple modules

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).

• It's clearly true if $I$ is finite, so we can restrict to $|I|\geq\aleph_0$. I have some things I think are true when $I$ is infinite, but it will take a better set-theorist than myself to justify it. The intuition I have is that if $|I|\leq |M_i|$, then $|\oplus_{i\in I}M_i|=|M_i|$ and that if $|I|> |M_i|$, $|\oplus_{i\in I}M_i|=|I|$. Please take this with a grain of salt, because I am not 100% sure both these facts are true. – rschwieb Mar 20 at 13:46
• I would assume it is, and I would proof it by testing it with a copy of $M_j \cong K_i$, since you know precisely the form of those morphism spaces ($k J$ and $k I$ where $k X$ is the free vectorspace on $X$). – Enkidu Mar 20 at 13:53
• If your ring is commutative, it follows from uniqueness of cardinality of bases over vector spaces, right ? @rschwieb : your cardinality computations are correct, in fact in general $|\oplus_i M_i| = \displaystyle\sum_n \sum_{E\subset I, |E| = n} |M_i|^n$, so when $|M_i|$ is finite this is just the cardinality of the set of finite subsets of $I$, which is just $|I|$, and $|M_i|$ is infinite, this is just $|I||M_i| = \max \{|I|, |M_i|\}$ – Max Mar 20 at 13:57
• We can get even more precise by noting that $\hom(M,M_i) = \hom (K_j, M_i)^J = \hom(M_i, M_i)^I$, so if $\kappa = |\hom (M_i, M_i)|$ then $\kappa^I = \kappa^J$. Of course exponentiation of cardinals is a terrible thing, but if $|I|\leq |M_i|$ and $\kappa$ is small enough this might bring some information – Max Mar 20 at 14:07
• Also note $|\hom(M_i, M)|= \sum_{E\subset I, \mathrm{finite}} |\hom (M_i, M_i^E)|= \sum_{E\subset I, \mathrm{finite}} \kappa^E = |I|$ if $\kappa$ is finite, $=\kappa |I|$ otherwise. So this is more precise : if $\kappa \leq |I|$ then we get $|I|=|J|$ too (morally, this can happen if $M_i$ has very few endomorphisms, so for instance if $Z(R)$ is very small) – Max Mar 20 at 14:13

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.