On YJ-injectivity and annihilators. (English) Zbl 1090.16002

Author’s summary: 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.


16D50 Injective modules, self-injective associative rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D25 Ideals in associative algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16L60 Quasi-Frobenius rings