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Rings With $u^n-1$ Nilpotent For Each Unit $u$

Peter Danchev, Arash Javan, Ahmad Moussavi

Abstract

We continue the study in-depth of the so-called $n$-UU rings for any $n\geq 1$, that were defined by the first-named author in Toyama Math. J. (2017) as those rings $R$ for which $u^n-1$ is always a nilpotent for every unit $u\in R$. Specifically, for any $n\geq 2$, we prove that a ring is strongly $n$-nil-clean if, and only if, it is simultaneously strongly $π$-regular and an $(n-1)$-UU ring. This somewhat extends results due to Diesl in J. Algebra (2013), Abyzov in Sib. Math. J. (2019) and Cui-Danchev in J. Algebra Appl. (2020). Moreover, our results somewhat improves the ones obtained by Ko$ş$an et al. in Hacettepe J. Math. Stat. (2020).

Rings With $u^n-1$ Nilpotent For Each Unit $u$

Abstract

We continue the study in-depth of the so-called -UU rings for any , that were defined by the first-named author in Toyama Math. J. (2017) as those rings for which is always a nilpotent for every unit . Specifically, for any , we prove that a ring is strongly -nil-clean if, and only if, it is simultaneously strongly -regular and an -UU ring. This somewhat extends results due to Diesl in J. Algebra (2013), Abyzov in Sib. Math. J. (2019) and Cui-Danchev in J. Algebra Appl. (2020). Moreover, our results somewhat improves the ones obtained by Koan et al. in Hacettepe J. Math. Stat. (2020).
Paper Structure (5 sections, 41 theorems, 56 equations, 1 table)

This paper contains 5 sections, 41 theorems, 56 equations, 1 table.

Table of Contents

  1. Introduction and Motivation
  2. Main Results and Examples
  3. Characterization properties
  4. Matrix rings and generalizations
  5. Group rings

Key Result

Proposition 2.1

Suppose $R$ is a ring of prime characteristic $p$ such that $m-1 | p-1$. Then, $R$ is an $(m-1)$-UU ring if, and only if, every invertible element of the ring R is $m$-strongly nil clean.

Theorems & Definitions (78)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • ...and 68 more