Commutatively of Prime and Semiprime ?-Rings with Symmetric BI-Derivations

Abstract

Let M be a ?-ring and let D: M x M ->M be a symmetric bi-derivation with the trace d: M -> M denoted by d(x) = D(x, x) for all x?M. The objective of this paper is to prove some results concerning symmetric bi-derivation on prime and semiprime ?-rings. If M is a 2-torsion free prime ?-ring and D ? 0 be a symmetric bi-derivation with the trace d having the property d(x)?x - x?d(x) = 0 for all x?M and ???, then M is commutative. We also prove another result in ?-rings setting analogous to that of Posner for prime rings.