I have an attempt to prove the claim that $R$ has finitely many nonisomorphic simple $R$-modules if $R$ is left artinian.
I would like to know if it's a good attempt. Helpful hints are very much welcome.
Attempt
Since $R$ is left artinian, $R$ has minimal left ideals. Also, $R$ is left semisimple, so $R$ can written as a direct sum of the minimal ideals, which can be grouped according to their isomorphic types as left $R-$modules. So, $R\cong n_{1}L_{1}\oplus\cdots\oplus n_{r}L_{r}$ where the $L_{r}$ are mutually nonisomorphic simple left $R-$modules. Let $N$ be any simple left $R-$module. Then $N\cong$a quotient of $R$ and thus $\cong L_{i}$ for some $i.$ Thus, $\left\{ L_{1},\cdots,L_{r}\right\} $ is the set of nonisomorphic left simple $R-$modules.