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.
Highlights
Throughout this note we assume that rings are associative may be without identity
The ring of real quaternions H is reversible with the natural involution ∗ defined on its elements by (a + bi + cj + dk)∗ = a − bi − cj − dk
First we have obtained a criterion for rings to be noninvolutary and we will find minimal right and left symmetric, symmetric, reversible, reflexive, noninvolutary, and ∗-reversible noncommutative rings
Summary
Throughout this note we assume that rings are associative may be without identity. We will mention if a ring is with the identity. The ring of real quaternions H is reversible with the natural involution ∗ defined on its elements by (a + bi + cj + dk)∗ = a − bi − cj − dk. One concludes that a ring with some involution ∗ may be commutative and reduced but not ∗-reversible.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
