Rings with xⁿ+ x or xⁿ- X nilpotent

Abstract

Let R be a ring and let n be an arbitrary but fixed positive integer. We characterize those rings R whose elements a satisfy at least one of the relations that an + a or an - a is a nilpotent whenever n ϵN\{1}. 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 xn - x nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with xn - x nilpotent, J. Algebra Appl. 21 (2022)], respectively.