One may decompose the right socle $S_r$ of any unital ring $R$ as $S_r=S_1\oplus S_2$, where $S_1$ is the sum of all nilpotent minimal right ideals, and $S_2$ is the sum of all idempotent minimal right ideals of $R$. I want to prove that $S_1$ and $S_2$ are two-sided ideals.
As for $S_1$ (resp. $S_2$) , let $x\in S_1$ (resp.$S_2$) and write $x=r_1+\cdots +r_n$, where $r_i\in I_i$, for some idempotent (resp. nilpotent) minimal right ideal $I_i$. If $s\in R$, it suffices to show that $sx\in S_1$(resp. $S_2$). Each $sr_i$ belongs to $sI_i$. But I do not know how to link these to expressions $xs$ and $r_is$. Incidentally, I do know that any sum of ifempotent right ideals is again an idempotent right ideal so that $S_2$ is such. Also, it is easily shown that $S_1^2=0$.
Thanks for any help!