Rings with xn + x or xn − x nilpotent

Abstract

Let [Formula: see text] be a ring and let [Formula: see text] be an arbitrary but fixed positive integer. We characterize those rings [Formula: see text] whose elements [Formula: see text] satisfy at least one of the relations that [Formula: see text] or [Formula: see text] is a nilpotent whenever [Formula: see text]. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with [Formula: see text] nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with [Formula: see text] nilpotent, J. Algebra Appl. 21 (2022)], respectively.