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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).

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  • $\begingroup$ 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. $\endgroup$ – rschwieb Mar 20 at 13:46
  • $\begingroup$ 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 $X$). $\endgroup$ – Enkidu Mar 20 at 13:53
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    $\begingroup$ 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 $|I||M_i| = \max \{|I|, |M_i|\}$ $\endgroup$ – Max Mar 20 at 13:57
  • $\begingroup$ 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 $\endgroup$ – Max Mar 20 at 14:07
  • $\begingroup$ 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) $\endgroup$ – Max Mar 20 at 14:13
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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”.

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  • $\begingroup$ Thank you. I have not been able to find the mentioned proof in Jacobson's book. Do you have a more precise reference inside the book? $\endgroup$ – Chris A Mar 24 at 13:34
  • $\begingroup$ @ChrisA Section 3.6 $\endgroup$ – egreg Mar 24 at 13:36

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