Weakly Clean Ideal

TL;DR

This paper introduces the concept of weakly clean ideals in ring theory, defining them based on elements decomposable into units and idempotents, and explores their properties.

Contribution

It formally defines weakly clean ideals and investigates their fundamental properties, expanding the theory of clean ideals in algebra.

Findings

Weakly clean ideals generalize clean ideals.

Characterization of weakly clean ideals in various rings.

Properties and examples of weakly clean ideals.

Abstract

Motivated by the concept of clean ideals, we introduce the notion of weakly clean ideals. We define an ideal $I$ of a ring $R$ to be weakly clean ideal if for any $x\in I$, $x=u+e$ or $x=u-e$, where $u$ is a unit in $R$ and $e$ is an idempotent in $R$. We discuss various properties of weakly clean ideals.