Lifting potent elements modulo nil ideals

Abstract

It is classically known that idempotents lift modulo nil one-sided ideals. So it is natural to ask if the same is true for potent elements. Although we answer this question in negative, we prove that the answer is positive in several special cases. For instance, the answer is positive in rings with finite characteristic. Let I be a nil one sided ideal of R and x ∈ R is such that x n + 1 − x ∈ I for some positive integer n . We prove that if n is a unit in R , then there exists an element p ∈ R with p n + 1 = p such that x − p ∈ I . By taking n = 1 , the result that idempotents lift modulo nil one sided ideals is retrieved. For a nil ideal I of an abelian ring R , we prove that potent elements lift modulo I precisely when torsion units or periodic elements lift modulo I . It follows that torsion units or periodic elements may also not lift modulo nil ideals. We prove that torsion units, potent elements and periodic elements lift modulo every nil ideal of a π -regular ring.