Let $M$ be a semisimple right module over the ring $R$, and let $T:= End_{R}(M)$, I want to prove that $M$ is a semisimple left module over the ring $T$. As $M$ is semisimple we have the following descomposition into simple modules
$$M=S_{1} \oplus S_{2} \oplus ....\oplus S_{n}$$.
I want to find a similar descomposition into simples for $M$ as a left module ove $T$. So far, I have succesfully proved that if $S$ is a simple right module over $R$, then $S$ is a left simple module over $End_{R}(S,S)$. But what I need to kill the original problem is to show that $S$ is left simple module over $T$ in order to use the original semisimple descomposition for $M$ as left $R$ module.
I have been trying this problem for a while and know it has been asked a few times but still see te explanations so foggy for being honest, also there is no even sketches of this proof online. So I would really aprreciate help proving this.
But you can't. It's not true. For a field $F$, $M=F\times F$ is a semisimple $F$ module, where both $F\times \{0\}$ and $\{0\}\times F$ are simple $F$ submodules of $M_F$, but neither one is an $End(M_F)\cong M_2(F)$ module, simple or not. But notice that both submodules are isomorphic as $F$ modules, and their sum is an $End(M_F)$ module: a simple one, in fact, showing $M$ is a semisimple $End(M_F)$ module.
That is part of the reason in your last post on this subject that I said you need to gather up the simple $R$ submodules that are $R$-isomorphic, because they form components which are the simple $T$ submodules you are looking for.