\(\sigma\)-derivations on prime near-rings (Q2752326)

Let \(N\) denote a zero-symmetric left near-ring and \(\sigma\) an automorphism of \(N\). An additive endomorphism \(D\) of \(N\) is called a \(\sigma\)-derivation if \(D(xy)=\sigma(x)D(y)+D(x)y\) for all \(x,y\in N\). This paper extends some commutativity results involving derivations, due to the reviewer and \textit{G. Mason} [Near-rings and near-fields, Proc. Conf., Tübingen/F.R.G. 1985, North-Holland Math. Stud. 137, 31-35 (1987; Zbl 0619.16024)]. A typical theorem reads as follows: If \(N\) is a 3-prime near-ring admitting a nontrivial \(\sigma\)-derivation \(D\) such that \(D(x)D(y)=D(y)D(x)\) for all \(x,y\in N\), then \((N,+)\) is Abelian. Moreover, if \(N\) is 2-torsion-free and \(\sigma\) and \(D\) commute, then \(N\) is a commutative ring.