\(\delta\)-derivations of prime alternative and Mal'tsev algebras (Q2729362)

Let \(\Phi\) be a ring, let \(\delta\) be a fixed element of \(\Phi\), and let \(A\) be a \(\Phi\)-algebra. A \(\Phi\)-linear mapping \(\varphi\) satisfying the equality \((xy)\varphi=\delta (x\varphi)y+ \delta x(y\varphi)\) is called a \(\delta\)-derivation of \(A\). In [Sib. Math. J. 40, No. 1, 174-184 (1999; Zbl 0936.17021)] the author proved that a prime Lie \(\Phi\)-algebra does not have a nonzero \(\delta\)-derivation if \(\delta \neq -1,0,\frac{1}{2},1\) and described the \(\delta\)-derivations of prime Lie \(\Phi\)-algebras for \(\delta=-1,\frac{1}{2}\). The elements of the centroid of an algebra (which are \(\frac{1}{2}\)-derivations) and the \(\delta\)-derivations for \(\delta=0,1\) are called trivial by the author. The author proves that the \(\delta\)-derivations of prime alternative (\(\frac{1}{6}\in\Phi\)) and non-Lie Mal'tsev (\(\frac{1}{2}\in\Phi\)) algebras are trivial.