A Note on GQ-injectivity

Abstract

The purpose of this note is to improve several known results on GQ-injective rings. We investigate in this paper the von Neumann regularity of left GQ-injective rings. We give an answer a question of Ming in the positive. Actually it is proved that if R is a left GQ-injective ring whose simple singular left R-modules are GP-injective then R is a von Neumann regular ring. Throughout this paper all rings are associative with identity and modules are unitary. We write J(R), Zl(R) and Zr(R) for the Jacobson radical, the left sin- gular ideal and the right singular ideal of a ring R, respectively. A left R-module M is called generalized quasi-injective (briefly GQ-injective) if for any left sub- module N which is isomorphic to a complement left submodule of M, every left R-homomorphism from N into M may be extended to an endomorphism of M. R is called a left GQ-injective ring if RR is GQ-injective. Left GQ-injective rings gen- eralize left continuous rings in the sense of Utumi (9). In (5) Ming proved that if R is left GQ-injective ring then J(R) = Zl(R) and R/J(R) is von Neumann regular. Recall that a left R-module M is left principally injective (briefly left P-injective) if, for any principal left ideal Ra of R, every left R-homomorphism from Ra into M extends to one from R into M. A left R-module M is called generalized left principally injective (briefly left GP-injective) if, for any 0 6 a 2 R, there exists a positive integer n such that a n 6 0 and any left R-homomorphism of Ra n into M extends to one from R into M. Note that GP-injective modules defined here are also called YJ-injective modules in (6), (8). R is said to be a left GP-injective ring if RR is GP-injective. Recently it is known that GP-injective rings need not be P-injective (2). For any element a in R, we denote l(a) and r(a) for the left annihilator and the right annihilator of {a}, respectively.