On a Result of Levitzki

Abstract

A well known result of Levitzki [2, Lemma 1.1] is the following:Theorem. Let R be a ring and U a non-zero one-sided ideal of R. Suppose that given a∈U, an = 0 for a fixed integer n ≥ 1; then R has a non-zero nilpotent ideal.The purpose of this note is to observe some additional results which are related to the above.Theorem 1. Let R be a ring with no non-zero nil ideals and U an ideal of R. Suppose that a∊R is such that for every x∊U, axn(x) = 0 where n(x) ≥ 1 depends on x; then aU= Ua = 0.