The nilpotent regular element problem

TL;DR

This paper investigates the properties of nilpotent regular elements in rings, demonstrating that such elements are not necessarily unit-regular in general rings, contrasting with their behavior in exchange rings.

Contribution

It introduces a normal form for adjoining an inner inverse and shows the non-unit-regularity of nilpotent regular elements in general rings, highlighting differences across ring classes.

Findings

Nilpotent regular elements are not always unit-regular in general rings.

In exchange rings, nilpotent regular elements are unit-regular.

The normal form aids in analyzing inner inverses in ring theory.

Abstract

We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular elements in exchange rings (a large class of rings), and for general rings when all powers of the nilpotent element $x$ are regular.