On Idempotent Reflexive Rings

We introduce in this paper the concept of idempotent reflexive right ideals and concern with rings containing an injective maximal right ideal. Some known results for reflexive rings and right HI-rings can be extended to idempotent reflexive rings. As applications, we are able to give a new characterization of regular right self-injective rings with nonzero socle and extend a known result for right weakly regular rings. Throughout this paper, R denotes an associative ring not necessarily with unity unless otherwise stated. A right ideal I is said to be reflexive (2) if aRb µ I implies bRa µ I for a;b 2 R. A ring R is called reflexive if 0 is a reflexive ideal. In this paper we define an idempotent reflexive right ideal which is a nontrivial generaliza- tion of a reflexive right ideal. Some known results of Mason (2) are extended. For an idempotent reflexive ring R with unity, we prove that if R contains an injective maximal right ideal, then R is right self-injective. As a byproduct of this result, we obtain a new characterization of regular right self-injective rings with nonzero socle. This characterization is then used to prove that an idempotent reflexive right HI-ring is semisimple Artinian. Consequently we extend nontrivially a result in (7). Moreover we show that if R is an idempotent reflexive ring with unity and every simple singular right R-module is p-injective then R is a right weakly regular ring. Definition 1. A right ideal I is called idempotent reflexive if aRe µ I if and only if eRa µ I for a;e = e 2 2 R. We say that R is an idempotent reflexive ring when 0 is an idempotent reflexive ideal.