Let $R$ be a unital ring with right socle $Soc(R_R)$ such that $R/Soc(R_R)$ is right weakly regular, i.e all whose right ideals are idempotent. Is it true that every right ideal $I$ of $R$ decomposes as the sum of an idempotent right ideal and a semi-simple right ideal (i.e., one falling into $Soc(R_R)$)?
My try is to consider the right ideal $(I+Soc(R_R))/Soc(R_R)$ about which one deduces $I+Soc(R_R)=I^2+Soc(R_R)$ due to the hypothesis. So, $I$ would be a subset of $I^2+Soc(R_R)$.
Any help and/or suggestion is welcome!