On rings close to regular and $p$-injectivity

The following results are proved for a ring A: (1) If A is a fully right idem- potent 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 di- rect sum of division rings iff A is a left 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/MA 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/LA 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 YJ-injective ring with maximum condition on annihilators.