Reversible Rings with Involutions and Some Minimalities

Abstract

In continuation of the recent developments on extended reversibilities on rings, we initiate here a study on reversible rings with involutions, or, in short, *-reversible rings. These rings are symmetric, reversible, reflexive, and semicommutative. In this note we will study some properties and examples of *-reversible rings. It is proved here that the polynomial rings of *-reversible rings may not be *-reversible. A criterion for rings which cannot adhere to any involution is developed and it is observed that a minimal noninvolutary ring is of order 4 and that a minimal noncommutative *-reversible ring is of order 16.