# Weakly $r$-clean rings and weakly $\star$-clean rings

## TL;DR

This paper introduces and explores the properties of weakly $r$-clean, weakly $g(x)$-$r$-clean, and weakly $igstar$-clean rings, generalizing the concept of weakly clean rings in ring theory.

## Contribution

It defines new classes of rings—weakly $r$-clean, weakly $g(x)$-$r$-clean, and weakly $igstar$-clean—and investigates their properties and relationships.

## Key findings

- Defined weakly $r$-clean rings and their properties.
- Generalized to weakly $g(x)$-$r$-clean rings with polynomial functions.
- Introduced weakly $igstar$-clean and $igstar$-$r$-clean rings and discussed their properties.

## Abstract

Motivated by the concept of weakly clean rings, we introduce the concept of weakly $r$-clean rings. We define an element $x$ of a ring $R$ as weakly $r$-clean if it can be expressed as $x=r+e$ or $x=r-e$ where $e$ is an idempotent and $r$ is a regular element of $R$. If all the elements of $R$ are weakly $r$-clean then $R$ is called a weakly $r$-clean ring. We discuss some of its properties in this article. Also we generalise this concept of weakly $r$-clean ring to weakly $g(x)$-$r$-clean ring, where $g(x) \in C(R)[x]$ and $C(R)$ is the centre of the ring $R$. Finally we introduce the concept of weakly $\star$-clean and $\star$-$r$-clean ring and discuss some of their properties.