Prime rings with annihilator conditions on power values of derivations on multilinear polynomials (Q1596457)

Let \(R\) be a prime \(K\)-algebra for \(K\) a commutative ring and let \(D\) be a nonzero derivation of \(R\). Various results in the literature have considered the situation when for some \(S\subseteq R\), the left annihilator of \(a\in R\) contains \(\{D(s)^m\mid s\in S\) and \(m\) is fixed\}. Here, the author extends these results by proving that if \(f\in K\{x_1,\dots,x_n\}\), the free algebra over \(K\), is a multilinear polynomial that is not central valued on \(R^n\), then \(aD(f(r_1,\dots,r_n))^m=0\) for some \(a\in R\), all \(r_j\in R\), and a fixed \(m\geq 1\), forces \(a=0\).