On commutativity in gamma-near-rings with symmetric bi-\((\sigma,\tau)\)-derivations. (Q2837874)

A \(\Gamma\)-nearring is a triple \((N,+,\Gamma)\) such that \((N,+)\) is a group; \(\Gamma\) is a set of binary operations on \(N\) such that for each \(\alpha\in\Gamma\), \((N,+,\alpha)\) is a nearring; and \(x\alpha(y\beta z)=(x\alpha y)\beta z\) for all \(x,y,z\in N\) and \(\alpha,\beta\in\Gamma\). If \(\sigma,\tau\) are automorphisms of \(N\) and \(D\colon N\times N\to N\) is a symmetric bi-additive map, \(D\) is called a symmetric bi-\((\sigma,\tau)\)-derivation if \(D(x\alpha y,z)=D(x,z)\alpha\sigma(y)+\tau(x)\alpha D(y,z)\) for all \(a\in\Gamma\) and all \(x,y\in N\).NEWLINENEWLINE The authors show that a 2-torsion free prime \(\Gamma\)-nearring with multiplicative center \(Z\) must be commutative if it admits a nonzero symmetric bi-\((\sigma,\tau)\)-derivation satisfying one of the following conditions: (i) \(\sigma=\tau=1\) and \(d(N)\subseteq Z\), where \(d\) is the trace of \(D\); (ii) \(D(N,N)\subseteq Z\); (iii) \(D(U,U)\subseteq Z\), where \(U\) is a nonzero semigroup ideal of \(N\).