On rings close to regular and $p$-injectivity

summary:The following results are proved for a ring $A$: (1) If $A$ is a fully right idempotent ring having a classical left quotient ring $Q$ which is right quasi-duo, then $Q$ is a strongly regular ring; (2) $A$ has a classical left quotient ring $Q$ which is a finite direct sum of division rings iff $A$ is a left $\operatorname{TC}$-ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let $A$ have the following properties: (a) each maximal left ideal of $A$ is either a two-sided ideal of $A$ or an injective left $A$-module; (b) for every maximal left ideal $M$ of $A$ which is a two-sided ideal, $A/M_A$ is flat. Then, $A$ is either strongly regular or left self-injective regular with non-zero socle; (4) $A$ is strongly regular iff $A$ is a semi-prime left or right quasi-duo ring such that for every essential left ideal $L$ of $A$ which is a two-sided ideal, $A/L_A$ is flat; (5) $A$ prime ring containing a reduced minimal left ideal must be a division ring; (6) A commutative ring is quasi-Frobenius iff it is a $\operatorname{YJ}$-injective ring with maximum condition on annihilators