Weak Nil Clean Ideal

TL;DR

This paper introduces the concept of weak nil clean ideals in rings, generalizing nil clean ideals, and explores their properties and relationship with weak nil clean rings.

Contribution

It defines weak nil clean ideals, investigates their properties, and examines their connection to weak nil clean rings, extending the theory of clean ideals.

Findings

Weak nil clean ideals are characterized by specific element decompositions.

The paper establishes properties and conditions for weak nil clean ideals.

Connections between weak nil clean ideals and weak nil clean rings are analyzed.

Abstract

As a generalization of nil clean ideal, we define weak nil clean ideal of a ring. An ideal $I$ of a ring $R$ is weak nil clean ideal if for any $x\in I$, either $x=e+n$ or $x=-e+n$, where $n$ is a nilpotent element and $e$ is an idempotent element of $R$. Some interesting properties of weak nil clean ideal and its relation with weak nil clean ring have been discussed.