Weakly $I$-clean rings

TL;DR

This paper introduces the concept of weakly I-clean rings, characterizes their properties, and explores conditions under which rings are weakly I-clean or uniquely weakly I-clean, including implications for idempotent elements and ring quotients.

Contribution

It defines weakly I-clean rings, provides characterizations, and studies their properties, extending the theory of clean rings with new conditions involving ideals and idempotents.

Findings

R is uniquely weakly I-clean iff R/I is semi boolean.

Idempotents can be lifted weakly modulo I under certain conditions.

Characterization of weakly J-clean rings.

Abstract

In this article, we introduce the concept of weakly $I$-clean ring, for any ideal $I$ of a ring $R$. We show that, for an ideal $I$ of a ring $R$, $R$ is uniquely weakly $I$-clean if and only if $R/I$ is semi boolean and idempotents can be lifted uniquely weakly modulo $I$ if and only if for each $a\in R$, there exists a central idempotent $e\in R$ such that either $a-e\in I$ or $a+e\in I$ and $I$ is idempotent free. As a corollary, we characterize weakly $J$-clean ring. Also we study various properties of weakly $I$-clean ring.