I have a question about the decomposition of semisimple modules.
Assume that $M$ is a left semisimple $R-$modules.
If $M=\oplus_{A}T_{\alpha}=\oplus_{B}S_{\beta}$, where $A$, $B$ are indexed set and $T_\alpha$, $S_\beta$ are simple $R$-modules.
I want to prove that $\oplus_{A}T_{\alpha}$ and $\oplus_{B}S_{\beta}$ are equivalent, i.e. there exists $\sigma :A\rightarrow B$ is a bijection such that $T_\alpha\cong S_{\sigma(\alpha)}$.
Here are my atempts:
Firstly, we can get $\mathrm{card}(A)=\mathrm{card}(B)$ if we use the similar method from the vector space.
(If $V$ is a vector space, $X$ and $Y$ are basis of $V$ respectively, then $\mathrm{card}(X)=\mathrm{card}(Y).$
Then if $A$ is finite, I can get it by the Jordan-Holder theorem.
But if $A$ is finite, from $\mathrm{card}(A)=\mathrm{card}(B)$, I can get a bijection $\phi$ from $A$ to $B$.
But this bijection maynot satisfy $T_\alpha\cong S_{\phi(\alpha)}$
How can I get the $\sigma$ I want?
Any help and references are greatly appreciated.
Thanks!