Principally Small Injective Rings

Abstract. A right ideal I of a ring R is small in case for every proper right ideal Kof R, K + I 6= R. A right R-module M is called PS-injective if every R-homomorphismf : aR ! M for every principally small right ideal aR can be extended to R ! M. A ringR is called right PS-injective if R is PS-injective as a right R-module. We develop, inthis article, PS-injectivity as a generalization of P-injectivity and small injectivity. Manycharacterizations of right PS-injective rings are studied. In light of these facts, we getseveral new properties of a right GPF ring and a semiprimitive ring in terms of rightPS-injectivity. Related examples are given as well. 1. IntroductionThroughout this paper, R is an associative ring with identity and all modulesare unitary. Let R be a ring. The Jacobson radical and nil radical of R are denotedby J(R) and Nil(R), respectively. The right singular ideal is denoted by Z(R R ),the socles are denoted by soc(R R ) and soc( R R). If X is a subset of R, the right(resp. left) annihilator of X in R is denoted by r