English

Rings Such That $u-1$ Lies In $J^{\#}(R)$ For Each Unit $u$

Rings and Algebras 2025-10-28 v1 Representation Theory

Abstract

We investigate the so-called {\it UJ#UJ^{\#} rings}, a new type of rings in which every unit can be written as 1+j1+j with jJ#(R)j\in J^{\#}(R). These rings were defined and studied by Saini-Udar in Czechoslovak Math. J. (2025) under the name {\it JU\sqrt{J}U rings}. (See \cite{SU}.) This class extends both the classes of UU and UJ rings, but also has its own special properties. In this study, we present some additional results about UJ#UJ^{\#} rings that supply those from \cite{SU} explaining their connections with Dedekind-finite, semi-potent and Boolean rings, respectively, as well as we give several characterizations in this direction. We also examine how these rings behave under common ring constructions and find conditions for group rings to be UJ#UJ^{\#}. Moreover, our establishments shed a clearer picture of how unit elements interact with radical-like parts of a ring.

Keywords

Cite

@article{arxiv.2510.22611,
  title  = {Rings Such That $u-1$ Lies In $J^{\#}(R)$ For Each Unit $u$},
  author = {Peter Danchev and Mina Doostalizadeh and Mehrdad Esfandiar and Omid Hasanzadeh},
  journal= {arXiv preprint arXiv:2510.22611},
  year   = {2025}
}

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17 pages