Quasi J-ideals of Commutative Rings

TL;DR

This paper introduces quasi J-ideals as a generalization of J-ideals in commutative rings, explores their properties, characterizes rings where all ideals are quasi J-ideals, and defines quasi presimplifiable rings as a broader class.

Contribution

It defines and studies quasi J-ideals, provides characterizations in special rings, and introduces quasi presimplifiable rings as a new generalization of presimplifiable rings.

Findings

Characterizations of quasi J-ideals in certain rings

Rings where all proper ideals are quasi J-ideals

Equivalence between ideals in the Jacobson radical and quasi presimplifiable quotients

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.