Posner's second theorem and an annihilator condition (Q2717049)

\textit{E. C. Posner} [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)] proved that if for a non-zero derivation \(d\) on a prime ring \(R\) the commutator \([d(x),x]=d(x)x-xd(x)\) is central for every \(x\in R\), then \(R\) is commutative. This result has numerous generalizations in many directions. In the paper under review the authors prove the following theorem. If \(R\) is a prime algebra over a commutative ring \(K\) of characteristic different from 2, \(d\) is a non-zero derivation of \(R\), and \(f(x_1,\ldots,x_n)\) is a non-central for \(R\) multilinear polynomial in the free algebra \(K\langle X\rangle\), then the left annihilator of the set \(\{[d(f(r_1,\dots,r_n)),f(r_1,\dots,r_n)]\mid r_i\in R\}\) is equal to 0. The proof is intricated but is based on simple ideas and involves the theory of differential and generalized identities.