Generalizations of Principally Injective Rings

Abstract

A ring R is said to be rightP-injective if every homomorphism of a principal right ideal to R is given by left multiplication by an element of R. This is equivalent to saying thatlr(a)=Ra for everya∈R, wherelandrare the left and right annihilators, respectively. We generalize this to only requiring that for each 0≠a∈R,lr(a) contains Ra as a direct summand. Such rings are called right AP-injective rings. Even more generally, if for each 0≠a∈R there exists ann0 withan≠0 such that Ran is not small inlr(an),R will be called a right QGP-injective ring. Among the results for right QGP-injective rings we are able to show that the radical is contained in the right singular ideal and is the singular ideal with a mild additional assumption. We show that the right socle is contained in the left socle for semiperfect right QGP-injective rings. We give a decomposition of a right QGP-injective ring, with one additional assumption, into a semisimple ring and a ring with square zero right socle. In the third section we explore, among other things, matrix rings which are AP-injective, giving necessary and sufficient conditions for a matrix ring to be an AP-injective ring. © 1998 Academic Press.