Generalized derivations in semiprime gamma rings. (Q417246)

Summary: Let \(M\) be a 2-torsion-free semiprime \(\Gamma\)-ring satisfying the condition \(a\alpha b\beta c=a\beta b\alpha c\) for all \(a,b,c\in M\), \(\alpha,\beta\in\Gamma\), and let \(D\colon M\to M\) be an additive mapping such that \(D(x\alpha x)=D(x)\alpha x+x\alpha d(x)\) for all \(x\in M\), \(\alpha\in\Gamma\) and for some derivation \(d\) of \(M\). We prove that \(D\) is a generalized derivation.