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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