Generalized inverses in Z[x]=(vx+x2)

Abstract

In this paper, we give a $\ast$-ring $\mathbb{Z}[x]/(vx+x^{2})$,
where $v \in \mathbb{Z}$ and $\ast$ is defined as $(a_{1}+a_{2}x)^{\ast}=a_{1}-va_{2}-a_{2}x$,
where $a_{1}, a_{2} \in \mathbb{Z}$.
Mainly, some classical generalized inverses are considered in this ring,
such as regular inverses, group inverses, Moore-Penrose inverses and so on.
Furthermore, it's proven that this ring is isomorphic to a special second-order matrix ring.