On Strongly \( J^{\#} \)-Clean Rings
arXiv:2508.17368
Abstract
We define and examine the class of {\it strongly \( J^{\#} \)-clean rings} consisting of those rings such that each element of is the sum of an idempotent from and an element from 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.
18 pages