A ring R is said to be right P-injective if every homomorphism of a principal right ideal to R is given by left multiplication by an element of R. This is ลฝ . equivalent to saying that lr a s Ra for every a g R, where l and r are the left and right annihilators, respectively. We generalize this to only requiring that for each ลฝ . 0 / a g 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 g R there exists an n n
ลฝ n . n ) 0 with a / 0 such that Ra is not small in lr a , 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. แฎ 1998 Academic Press R * The research was supported by NSERC Grants A80204 and OGP0194196. 706