$\sqrt{J}$-clean rings
Dinesh Udar, Shiksha Saini
TL;DR
The paper introduces $\sqrt{J}$-clean rings, where every element splits as an idempotent plus an element of $\sqrt{J(R)}$, and establishes a key characterization: $R$ is $\sqrt{J}$-clean iff $R/J(R)$ is $\sqrt{J}$-clean and idempotents lift modulo $J(R)$. It situates this class between semiboolean/nil-clean and clean rings, proves that $\sqrt{J}$-clean rings are always clean, and explores strong and unique variants, including classifications for division, local, and semisimple cases. The study extends to matrix-ring and Morita-context constructions, showing when these extensions preserve the $\sqrt{J}$-clean property, and identifies the division-ring case $M_n(D)$ being $\sqrt{J}$-clean only when $D\cong\mathbb{F}_2$. Overall, the work provides a cohesive framework linking classic decompositions with lifting properties and matrix-analytic constructions.
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.