**Roger Yue Chi Ming**

## On YJ-Injectivity and Annihilators

**Abstract:**

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.

**Keywords:**

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

**MSC 2000:** 16D40, 16D50, 16E50