Extensions of strongly \alpha-reversible rings

Document Type: Research Paper


Nanjing University


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.