On Yaqub nil-clean ring

TL;DR

This paper characterizes Yaqub nil-clean rings, showing they are rings where each element differs from a cube root element by a nilpotent element, with a specific commutativity condition.

Contribution

It provides a necessary and sufficient condition for rings to be Yaqub nil-clean based on the existence of a special idempotent element.

Findings

Ring R is Yaqub nil-clean iff for each a, there exists e with e^3=e, such that a-e or a+3e is nilpotent and ae=ea.

The characterization involves the existence of a tripotent element e that commutes with all elements of R.

The paper establishes an equivalence condition for Yaqub nil-clean rings based on nilpotent and tripotent elements.

Abstract

A ring R is Yaqub nil-clean if for any a\in R, a-a^3 or a+a^3 is nilpotent for all a\in R. We prove that a ring R is Yaqub nil-clean if and only if for any a\in R, there exists some e^3=e\in R, such that a-e or a+3e is nilpotent and ae=ea.