Symmetric Rings with Involutions

Abstract

Two fairly useful notions to support some commutativity conditions for non commutative rings are symmetry and reversibility. Our aim in this note is to study *- symmetric rings, where * is an involution on the ring. A ring R with involution * is called *- symmetric if for any elements a,b,c∈R, abc=0 ⇒ acb*=0. Every *- symmetric ring with 1 is symmetric but the converse need not be true in general, even for the commutative rings. We discussed some characterizations in which these two notions and the notions of reversibility and *- reversibility coincide. We have extended *- symmetric rings to factor polynomial rings that are isomorphic to rings of Barnett matrices.