ELEMENTS IN EXCHANGE QB∞-RINGS

Abstract

An element $u\in R$ is pseudo invertible if there exist $v,w\in R$ such that $R(1-uv)R(1-wu)R$ is a nilpotent ideal. A ring $R$ is a $QB_{\infty}$ ring provided whenever $aR+bR=R$ with $a,b\in R$, there exists $y\in R$ such that $a+by$ is pseudo invertible. We prove, in this paper, that an exchange ring $R$ is a $QB_{\infty}$-ring if and only if whenever $x=xyx$, there exists a pseudo invertible $u\in R$ such that $x=xyu=uyx$ if and only if whenever $x=xyx$, there exists $a\in R$ such that $y+a$ is pseudo invertible and $1+xa$ is invertible. Also we characterize exchange $QB_{\infty}$-rings by virtue of pseudo unit-regularity. These generalize the main results of Wei (2004, Theorem 3, Theorem 7; 2005, Theorem 2.2, Theorem 2.4 and Theorem 3.6).