Roger Yue Chi Ming

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.

Annihilator, Noetherian, Artinian, quasi-Frobenius ring, p-injectivity, YJ-injectivity, von Neumann regular.

MSC 2000: 16D40, 16D50, 16E50