Extensions of strongly alpha-reversible rings

Abstract

We introduce the notion of<br />strongly $alpha$-reversible rings which is a strong version of<br />$alpha$-reversible rings, and investigate its properties. We first<br />give an example to show that strongly reversible rings need not be<br />strongly $alpha$-reversible. We next argue about the strong<br />$alpha$-reversibility of some kinds of extensions. A number of<br />properties of this version are established. It is shown that a ring<br />$R$ is strongly right $alpha$-reversible if and only if its<br />polynomial ring $R[x]$ is strongly right $alpha$-reversible if and<br />only if its Laurent polynomial ring $R[x, x^{-1}]$ is strongly right<br />$alpha$-reversible. Moreover, we introduce the concept of<br />Nil-$alpha$-reversible rings to investigate the nilpotent elements<br />in $alpha$-reversible rings. Examples are given to show that right<br />Nil-$alpha$-reversible rings need not be right $alpha$-reversible.