GENERALIZATIONS OF PRINCIPALLY INJECTIVE-RINGS

A ring R is said to be right P-injective if every homomorphism of a pr incipal right ideal to R is given by left multiplication by an element of R. This is equivalent to saying that Ir(a) = Ra for every a epsilo n R, where 1 and r are the left and right annihilators, respectively. We generalize this to only requiring that for each 0 not equal a epsil on R, Ir(a) contains Ra as a direct summand. Such rings are called rig ht AP-injective rings. Even more generally, if for each 0 not equal a epsilon R there exists an n > 0 with a(n) not equal 0 such that Ra-n i s not small in Ir(a(n)), R will be called a right QGP-injective ring. Among the results for right QGP-injective rings we are able to show th at the radical is contained in the right singular ideal and is the sin gular ideal with a mild additional assumption. We show that the right socle is contained in the left socle for semiperfect right QGP-injecti ve rings. We give a decomposition of a right QGP-injective ring, with one additional assumption, into a semisimple ring and a ring with squa re zero right socle. In the third section we explore, among other thin gs, matrix rings which are AP-injective, giving necessary and sufficie nt conditions for a matrix ring to be an AP-injective ring. (C) 1998 A cademic Press.