On Strongly \( J^{\#} \)-Clean Rings

TL;DR

This paper introduces and studies strongly J#-clean rings, showing they are strongly clean, Dedekind-finite, and have Boolean factor rings, and establishes their equivalence with strongly J-clean rings, with extensions to group and matrix rings.

Contribution

It defines strongly J#-clean rings, proves their properties, and shows they coincide with strongly J-clean rings, extending the concept to group and matrix rings.

Findings

Strongly J#-clean rings are strongly clean and Dedekind-finite.

Their factor rings modulo the Jacobson radical are Boolean.

The classes of strongly J#-clean and strongly J-clean rings are equivalent.

Abstract

We define and examine the class of {\it strongly \( J^{\#} \)-clean rings} consisting of those rings $R$ such that each element of $R$ is the sum of an idempotent from $R$ and an element from $J^{\#}(R)$ that commute with each other. More exactly, we prove that these rings are simultaneously strongly clean and Dedekind-finite as well as that they factor-ring modulo the Jacobson radical is always Boolean, and also provide some close relations with certain other well-established classes of rings like these of local, semi-local and strongly J-clean rings (as introduced by Chen on 2010) showing the surprising fact that the classes of strongly \( J^{\#} \)-clean and strongly J-clean rings, actually, do coincide. Moreover, a few more extensions of the newly defined class such as group rings and generalized matrix rings are provided too.