On two questions of Nicholson

TL;DR

This paper characterizes rings with stable range one through properties of units and principal ideals, and confirms Nicholson's questions about the conditions for stable range one in specific ring classes.

Contribution

It provides necessary and sufficient conditions for stable range one in rings, answering Nicholson's recent questions.

Findings

A ring has stable range one iff every left unit lifts modulo every left principal ideal.

A left quasi-morphic ring has stable range one iff it is left uniquely generated.

The paper affirms Nicholson's two questions regarding stable range one.

Abstract

We show that a ring R has stable range one if and only if every left unit lifts modulo every left principal ideal. We also show that a left quasi-morphic ring has stable range one if and only if it is left uniquely generated. Thus we answer in the affirmative the two questions raised recently by W. K. Nicholson.