For readers' benefit, a few definitions for a ring $R$.
The left (right) socle of $R$ is the sum of all minimal left (right) ideals of $R$. It may happen that it is zero if no minimals exist.
A ring is semiperfect if all finitely generated modules have projective covers.
Is there a semiperfect ring with zero left socle and nonzero right socle?
Someone asked me this recently, and nothing sprang to mind either way. In any case, I'm interested in a construction method that is amenable for creating imbalanced socles like this.
If you happen to know the answer when semiperfect is strengthened to be 'some side perfect' or 'semiprimary' or 'some side Artinian', then please include it as a comment. (Of course, a ring will have a nonzero socle on a side on which it is Artinian.)