Quasi J-ideals of commutative rings

Abstract

Let R be a commutative ring with identity. In this paper, we introduce the concept of quasi J-ideal which is a generalization of J-ideal. A proper ideal of R is called a quasi J-ideal if its radical is a J-ideal. Many characterizations of quasi J-ideals in some special rings are obtained. We characterize rings in which every proper ideal is quasi J-ideal. Further, as a generalization of presimplifiable rings, we define the notion of quasi presimplifiable rings. We call a ring R a quasi presimplifiable ring if whenever \(a,b\in R\) and \(a=ab\), then either a is a nilpotent or b is a unit. It is shown that a proper ideal I that is contained in the Jacobson radical is a quasi J-ideal (resp. J-ideal) if and only if R/I is a quasi presimplifiable (resp. presimplifiable) ring.