On injectivity and $p$-injectivity

This note contains the following results for a ring A : (1) A is simple Artinian if and only if A is a prime right Y J- injective, right and left V -ring with a maximal right annihilator ; (2) if A is a left quasi-duo ring with Jacobson radical J such that AA=J is p-injective, then the ring A=J is stongly regular ; (3) A is von Neumann regular with non-zero socle if and only if A is a left p:p:ring containing a finitely generated p-injective maximal left ideal satisfying the following condition : if e is an idempotent in A, then eA is a minimal right ideal if and only if Ae is a minimal left ideal ; (4) If A is left non-singular, left Y J-injective such that each maximal left ideal of A is either injective or a two-sided ideal of A, then A is either left self-injective regular or strongly regular ; (5) A is left continuous regular if and only if A is right p-injective such that for every cyclic left A-module M, AM=Z(M) is projective. ((5) remains valid if ?continuous? is replaced by ?self-injective? and ?cyclic? is replaced by ?finitely generated?). Finally, we have the following two equivalent properties for A to be von Neumann regular : (a) A is left non-singular such that every finitely gener- ated left ideal is the left annihilator of an element of A and every principal right ideal of A is the right annihilator of an element of A ; (b) Change ?left non-singular? into ?right non-singular? in (a).