If $P$ is a module over the semisimple ring $R/J$, where $R$ is a semilocal ring having $1$, and $J$ is its Jacobson radical, does any isomorphism $P⊕...⊕P≅P'⊕...⊕P'$ with the same (finite) number of components each side lead to $P≅P'$? Any cooperation would be appreciated.
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There's no need to consider $J$; just assume $R$ is semisimple. Every (right) module over a semisimple ring is a direct sum of simple modules. Moreover, consider $(S_i)_{i\in I}$, a family of simple modules over $R$ such that
- every simple $R$-module $S$ is isomorphic to $S_i$, for some $i\in I$;
- if $i\ne j$, then $S_i$ is not isomorphic to $S_j$.
We can decompose every module $M$ as $$ M=\bigoplus_{i\in I} S_i^{(\alpha_i)} $$ where $N^{(\alpha)}$ means the direct sum of $\alpha$ (a cardinal) copies of $N$.
Then we can prove that $$ \bigoplus_{i\in I} S_i^{(\alpha_i)}\cong\bigoplus_{i\in I} S_i^{(\beta_i)} $$ if and only if $\alpha_i=\beta_i$, for all $i\in I$. This is a particular case of theorem 12.4 in Anderson-Fuller “Rings and Categories of Modules”.
Now, if $\alpha_i$ is infinite and $n>0$ is an integer, then $(S_i^{(\alpha_i)})^n\cong S_i^{(\alpha_i)}$; if $\alpha_i$ is finite, $(S_i^{(\alpha_i)})^n\cong S_i^{(n\alpha_i)}$. Thus your claim follows.
Alternatively one could use the more general Krull-Remak-Schmidt-Azumaya theorem that you find in these notes as Theorem 2.12.
