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A module $M$ over a ring $R$ is called semisimple if it admits a direct sum decomposition into simple modules. If $M$ admits a finite decomposition $M=\bigoplus_{i=1}^n S_i$ into simple $R$-modules $S_i$, then this decomposition is unique up to isomorphism and permutation of the factors because from such a decomposition one can cook up a composition series and then use the Jordan–Hölder theorem.

I would like to know: are infinite decompositions unique as well?

E.g. if $R$ is semisimple, then all $R$-modules are semisimple. Do they decompose uniquely?

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    $\begingroup$ What's unique is the decomposition is into isotypic summands. The decomposition into simple summands is only unique up to isomorphism. $\endgroup$
    – YCor
    Dec 13 '20 at 14:14
  • $\begingroup$ Yes, see Decomposition of a module — this is Azumaya theorem. $\endgroup$
    – abx
    Dec 13 '20 at 14:15
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    $\begingroup$ A reference is T-Y. Lam. “Lectures on modules and rings”. Graduate Texts in Math. 189. SpringerVerlag, New York, 1999. $\endgroup$
    – YCor
    Dec 13 '20 at 14:15
  • $\begingroup$ @abx: You sure? This is about the Krull-Schmidt theorem which I thought only applies to finite direct sum decompositions!? $\endgroup$
    – user78294
    Dec 13 '20 at 14:36
  • $\begingroup$ @YCor: Sorry, could you elaborate a bit more? I'm not sure I got this. $\endgroup$
    – user78294
    Dec 13 '20 at 14:37
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Yes. A semisimple module $M$ is canonically isomorphic to

$$M \cong \bigoplus_i \text{Hom}_R(S_i, M) \otimes_{\text{End}(S_i)} S_i$$

where $\text{Hom}_R(S_i, M)$ is what you might call the multiplicity space of the simple module $S_i$. It is naturally a module over the division ring $\text{End}(S_i)$, and from this expression it follows that the multiplicity of $S_i$ in any direct sum decomposition of $M$ into simples must be $\dim_{\text{End}(S_i)} \text{Hom}_R(S_i, M)$. (For this we need to know that $\text{Hom}_R(S_i, -)$ preserves infinite coproducts, but this follows from the fact that simple modules are cyclic and so in particular finitely generated.)

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