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Authors:
Title:
Rings such that, for each unit $u, u^n - 1$ belongs to the $\Delta(R)$
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Discussiones Mathematicae - General Algebra and Applications
Received: 2024-11-01 , Revised: 2024-12-10 , Accepted: 2024-12-10 , Available online: 2025-07-30 , https://doi.org/10.7151/dmgaa.1483
Abstract:
We study in-depth those rings R for which, there exists a fi xed 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)-\DeltaU 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)-\DeltaU rings. These achievements considerably extend results established by Danchev (Rend. Sem. Mat. Univ. Pol. Torino, 2019) and Kosan et al. (Hacettepe J. Math. & Stat., 2020). Some other closely related results of this branch are also established.
Keywords:
$n$-$\Delta$U ring, $\Delta$U ring, $n$-JU ring, JU ring, (semi-)regular ring, clean ring
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