A sufficient condition is given for a ring to be either strongly regular or left self-injective regular with non-zero socle. If $A$ is a left self-injective ring such that the left annihilator of each element is a cyclic flat left $A$-module, then $A$ is left self-injective regular. Quasi-Frobenius rings are characterized. A right non-singular, right YJ-injective right FPF ring is left and right self-injective regular of bounded index. Right YJ-injective strongly $\pi$-regular rings have nil Jacobson radical. P.I.-rings whose essential right ideals are idempotent must be strongly $\pi$-regular. If every essential left ideal of $A$ is an essential right ideal and every singular right $A$-module is injective, then $A$ is von Neumann regular, right hereditary.