Socle of a ring $R$

Question

It is well-known that for an idempotent $e\in R$, the right $R$-module $eR$ is simple faithful if and only if $Re$ is a simple faithful left $R$-module. Now, I want to prove that when $Re$ is simple faithful $ReR$ is a submodule of the right socle of $R_R$. To this end, we must show that it is a submodule of the sum of all simple right submodules of $R_R$. Any help?

Answer 1

If $eR$ is simple, then $reR$ is either $0$ or a simple submodule of $R_R$, because left multiplication by $r$ is an endomorphism of $R_R$.

More generally, the socle of $R_R$ is a two sided ideal, because, for any simple submodule $S$ of $R_R$ and any $r\in R$, $rS=0$ or $rS$ is simple.