Rings such that, for each unit $u$, $u^n-1$ belongs to the $\Delta(R)$

TL;DR

This paper characterizes rings where each unit minus one raised to a fixed power lies in a specific subring, extending previous results and linking properties like exchange and cleanness in these rings.

Contribution

It provides a complete description of reduced -regular - rings satisfying the -1 property for units, and establishes the equivalence of exchange and clean properties in these rings.

Findings

Rings satisfying the property are characterized by the equation x^{2n}=x.

The property of being exchange and clean are equivalent in these rings.

The results extend previous work by Danchev and Kofan et al.

Abstract

We study in-depth those rings $R$ for which, there exists a fixed $n\geq 1$, such that $u^n-1$ lies in the subring $\Delta(R)$ of $R$ for every unit $u\in R$. We succeeded to describe for any $n\geq 1$ all reduced $\pi$-regular $(2n-1)$-$\Delta$U rings by showing that they satisfy the equation $x^{2n}=x$ as well as to prove that the property of being exchange and clean are tantamount in the class of $(2n-1)$-$\Delta$U rings. These achievements considerably extend results established by Danchev (Rend. Sem. Mat. Univ. Pol. Torino, 2019) and Ko\c{s}an et al. (Hacettepe J. Math. \& Stat., 2020). Some other closely related results of this branch are also established.