The \(k\)-derivation of a gamma-ring (Q2703113)

Let \(M\) and \(\Gamma\) be two additive Abelian groups and let there exist a mapping \(M\times\Gamma\times M\) to \(M\) such that for all \(a,b,c\in M\) and \(\alpha,\beta\in\Gamma\) 1. \((a+b)\alpha c=a\alpha c+b\alpha c\), 2. \(a(\alpha+\beta)b=a\alpha b+a\beta b\), 3. \(a\alpha(b+c)=a\alpha b+a\alpha c\), 4. \((a\alpha b)\beta c=a\alpha(b\beta c)\). Here \(a\alpha b\) denotes the image of \((a,\alpha,b)\). In this case \(M\) is called gamma-ring. Let \(k\) be an additive mapping from \(\Gamma\) to \(\Gamma\). An additive mapping \(d\colon M\to M\) is called \(k\)-derivation of \(M\) if for all \(a,b\in M\), \(\alpha\in\Gamma\) \(d(a\alpha b)=d(a)\alpha b+ak(\alpha)b+a\alpha d(b)\). In the paper some conditions when a gamma-ring is commutative or when a \(k\)-derivation is an ordinary one are given.