Extensions of strongly alpha-reversible rings

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