Symmetric Rings with Involutions
Wafaa Mohammed Fakieh *
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.
*Author to whom correspondence should be addressed.
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.
Keywords: *-symmetric rings, *-reversible rings, *-rigid rings