$\sqrt{J}$-clean rings

TL;DR

This paper introduces and studies the properties of $\,\sqrt{J}$-clean rings, a new class of rings where each element decomposes into an idempotent and an element from the radical's root, expanding understanding of ring decompositions.

Contribution

It defines $\,\sqrt{J}$-clean rings, explores their properties, and characterizes them, also relating them to existing classes like clean, semiboolean, and nil clean rings, with matrix extension analysis.

Findings

$\,\sqrt{J}$-clean rings generalize clean rings.

Characterization of $\,\sqrt{J}$-clean rings via quotient and lifting properties.

Matrix extensions of $\,\sqrt{J}$-clean rings are studied.

Abstract

In this paper, we study a new class of rings, called $\sqrt{J}$-clean rings. A ring in which every element can be expressed as the addition of an idempotent and an element from $\sqrt{J(R)}$ is called a $\sqrt{J}$-clean ring. Here, $\sqrt{J(R)}=\{ z\in R : z^n\in J(R) \ \mathrm{for \ some} \ n \geq 1 \}$ where, $J(R)$ is the Jacobson radical. We provide the basic properties of $\sqrt{J}$-clean rings. We also show that the class of semiboolean and nil clean rings is a proper subclass of the class of $\sqrt{J}$-clean rings, which itself is a proper subclass of clean rings. We obtain basic properties of $\sqrt{J}$-clean rings and give a characterization of $\sqrt{J}$-clean rings: a ring $R$ is a $\sqrt{J}$-clean ring iff $R/J(R)$ is a $\sqrt{J}$-clean ring and idempotents lift modulo $J(R)$. We also prove that a ring is a uniquely clean ring if and only if it is a uniquely $\sqrt{J}$-clean ring. Finally, several matrix extensions like $T_n(R)$ and $D_n(R)$ over a $\sqrt{J}$-clean ring are explored.