On trace of symmetric of bi-gamma-derivations in gamma-near-rings

Abstract

Let M be a 2-torsion free 3-prime left ?-near-ring with multiplicative center C. For x ? M, let C(x) be the centralizer of x in M. The aim of this paper is to study the trace of symmetric bi-?-derivations (also symmetric bi-generalized ?-derivations) on M. Main results are the following theorems: Let D(.,.) be a non-zero symmetric bi-?-derivation of M and F(.,.) a symmetric bi-additive mapping of M. Let d and f be traces of D(.,.) and F(.,.), respectively. In this case (1) If d(M) ? C, then M is a commutative ring. (2) If d(y), d(y) + d(y) ? C(D(x, z)) for all x, y, z ? M, then M is a commutative ring. (3) If F(.,.) is a non-zero symmetric bi-generalized ?-derivation of M associated with D(.,.) and f(M) ? C, then M is a commutative ring. (4 ) If F(.,.) is a non-zero symmetric bi-generalized ?-derivation of M associated with D(.,.) and f(y), f(y) + f(y) ? C(D(x, z)) for all x, y, z ? M, then M is a commutative ring. © 2007 University of Houston.