On YJ-Injectivity and Annihilators

This note contains the following results for a ring A : (1) A is a quasi-Frobenius ring iff A is a left and right YJ-injective, left Noetherian ring whose prime factor rings are right YJ-injective iff every non-zero one-sided ideal of A is the annihilator of a finite subset of elements of A ; (2) if A is a right YJ-injective ring such that any finitely generated right ideal is either a maximal right annihilator or a projective right annihilator, then A is either quasi-Frobenius or a right p.p. ring such that every non-zero left ideal of A contains a non-zero idempotent; (3) a commutative YJ-injective Goldie ring is quasi-Frobenius; (4) if the Jacobson radical of A is reduced, every simple left A -module is either YJ-injective or flat and every maximal left ideal of A is either injective or a two-sided ideal of A , then A is either strongly regular or left self-injective regular with non-zero socle.