A Class of $J$-quasipolar Rings

TL;DR

This paper introduces and studies weakly J-quasipolar rings, a new class of rings that generalize J-quasipolar rings, providing characterizations and exploring their properties and relationships with other ring classes.

Contribution

It defines weakly J-quasipolar rings, establishes their properties, and shows they form a class between J-quasipolar and quasipolar rings, connecting to uniquely clean rings.

Findings

R/J(R) is weakly J-quasipolar and commutative.

R/J(R) is reduced.

Weakly J-quasipolar rings lie between J-quasipolar and quasipolar rings.

Abstract

In this paper, we introduce a class of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a$ of a ring $R$ is called {\it weakly $J$-quasipolar} if there exists $p^2 = p\in comm^2(a)$ such that $a + p$ or $a-p$ are contained in $J(R)$ and the ring $R$ is called {\it weakly $J$-quasipolar} if every element of $R$ is weakly $J$-quasipolar. We give many characterizations and investigate general properties of weakly $J$-quasipolar rings. If $R$ is a weakly $J$-quasipolar ring, then we show that (1) $R/J(R)$ is weakly $J$-quasipolar, (2) $R/J(R)$ is commutative, (3) $R/J(R)$ is reduced. We use weakly $J$-quasipolar rings to obtain more results for $J$-quasipolar rings. We prove that the class of weakly $J$-quasipolar rings lies between the class of $J$-quasipolar rings and the class of quasipolar rings. Among others it is shown that a ring $R$ is abelian weakly $J$-quasipolar if and only if $R$ is uniquely clean.