Generalizations of Principally Injective Rings,

A ringRis said to be rightP-injective if every homomorphism of a principal right ideal toRis given by left multiplication by an element ofR. This is equivalent to saying thatlr(a) = Rafor everya R, wherelandrare the left and right annihilators, respectively. We generalize this to only requiring that for each 0 ≠ a R,lr(a) containsRaas a direct summand. Such rings are called rightAP-injective rings. Even more generally, if for each 0 ≠ a Rthere exists ann > 0 withan ≠ 0 such thatRanis not small inlr(an),Rwill be called a rightQGP-injective ring. Among the results for rightQGP-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 rightQGP-injective rings. We give a decomposition of a rightQGP-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 areAP-injective, giving necessary and sufficient conditions for a matrix ring to be anAP-injective ring.