On Perfectly Clean Rings

TL;DR

This paper explores conditions under which certain matrix rings over local rings are perfectly clean, revealing that perfect cleanness coincides with strong cleanness in these contexts and providing criteria for perfect J-cleanness.

Contribution

It establishes new conditions for perfect cleanness in matrix and triangular rings, showing equivalence with strong cleanness and offering criteria for perfect J-cleanness.

Findings

Perfect cleanness and strong cleanness coincide in matrix rings over local rings.

Criteria for triangular matrix rings to be perfectly J-clean are established.

For commutative rings, matrix rings are perfectly J-clean if and only if the base ring is strongly J-clean.

Abstract

An element $a$ of a ring $R$ is called perfectly clean if there exists an idempotent $e\in comm^2(a)$ such that $a-e\in U(R)$. A ring $R$ is perfectly clean in case every element in $R$ is perfectly clean. In this paper, we investigate conditions on a local ring $R$ that imply that $2\times 2$ matrix rings and triangular matrix rings are perfectly clean. We shall show that for these rings perfect cleanness and strong cleanness coincide with each other, and enhance many known results. We also obtain several criteria for such a triangular matrix ring to be perfectly $J$-clean. For instance, it is proved that for a commutative ring $R$, $T_n(R)$ is perfectly $J$-clean if and only if $R$ is strongly $J$-clean.