On the centroid of the prime gamma rings. (Q2706521)

Defninition. The set \(C_\Gamma:=\{g\in Q\mid g\gamma f=f\gamma g\) for all \(f\in Q\) and \(\gamma\in\Gamma\}\) is called the extended centroid of a \(\Gamma\)-ring \(M\), where \(Q\) denotes the quotient \(\Gamma\)-ring of \(M\).NEWLINENEWLINENEWLINEMain result: Theorem. Let \(M\) be a 2-torsion free prime \(\Gamma\)-ring, \(D_1(\cdot,\cdot)\), \(D_2(\cdot,\cdot)\), \(D_3(\cdot,\cdot)\) and \(D_4(\cdot,\cdot)\) symmetric bi-derivations of \(M\) and \(d_1\), \(d_2\), \(d_3\) and \(d_4\) the traces of \(D_1(\cdot,\cdot)\), \(D_2(\cdot,\cdot)\), \(D_3(\cdot,\cdot)\) and \(D_4(\cdot,\cdot)\), respectively. If \(d_1(x)\gamma d_2(y)=d_3(x)\gamma d_4(y)\) for all \(x,y\in M\) and \(\gamma\in\Gamma\) and \(d_1\neq 0\neq d_4\), then there exists \(\lambda\in C_\Gamma\) such that \(d_2(x)=\lambda\alpha d_4(x)\) and \(d_3(x)=\lambda\alpha d_1(x)\) for \(\alpha\in\Gamma\) where \(C_\Gamma\) is the extended centroid of \(M\).