A morphic ring of neat range one

Abstract

We show that a commutative ring \(R\) has neat range one if and only if every unit modulo principal ideal of a ring lifts to a neat element. We also show that a commutative morphic ring \(R\) has a neat range one if and only if for any elements \(a, b \in R\) such that \(aR=bR\) there exist neat elements \(s, t \in R\) such that \(bs=c\), \(ct=b\). Examples of morphic rings of neat range one are given.