Resumen de On Trace of Symmetric Bi-Gamma-Derivations in Gamma-Near-Rings

Mustafa Uçkun, Mehmet Ali Öztürk

• Let M be a 2-torsion free 3-prime left gamma-near-ring with multiplicative center C. Let x be an element of M and C(x) the centralizer of x in M. The aim of this paper is to study the trace of symmetric bi-gamma-derivations (also symmetric bi-generalized gamma-derivations) on M. Main results are the following theorems: Let D(.,.) be a non-zero symmetric bi-gamma-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) is a subset of C, then M is a commutative ring. (2) If d(y), d(y) + d(y) are elements of C(D(x,z)) for all x, y, z in M, then M is a commutative ring. (3) If F(.,.) is a non-zero symmetric bi-generalized gamma-derivation of M associated with D(.,.) and f(M) is a subset of C, then M is a commutative ring. (4) If F(.,.) is a non-zero symmetric bi-generalized gamma-derivation of M associated with D(.,.) and f(y), f(y) + f(y) are elements of C(D(x,z)) for all x, y, z in M, then M is a commutative ring.