Quasi-unit regularity and $QB$-rings

Abstract

Some relations for quasiunit regular rings and $QB$-rings, as well as for pseudounit regular rings and $QB_{\infty}$-rings, are obtained. In the first part of the paper, we prove that (an exchange ring $R$ is a $QB$-ring) (whenever $x \in R$ is regular, there exists a quasiunit regular element $w \in R$ such that $x = xyx = xyw$ for some $y \in R$) — (whenever $aR + bR = dR$ in $R$, there exists a quasiunit regular element $w \in R$ such that $a + bz = dw$ for some $z \in R$). Similarly, we also give necessary and sufficient conditions for $QB_{\infty}$-rings in the second part of the paper.