Power values of derivations on multilinear polynomials in prime rings. (Q255683)

Let \(R\) be a prime ring with center \(Z\), nonzero ideal \(I\), fixed integers \(m,p\geq 1\), extended centroid \(C\), nonzero derivation \(d\), and \(f\in C\{x_1,\ldots,x_n\}\), nonzero and multilinear. Consider \(F([x_1,\ldots,x_n])=((d(f(x_1,\ldots,x_n)))^m-f(x_1,\ldots,x_n))^p\colon I^n\to IC\) by evaluation. The main results of the authors are: if \(F(I^n)=0\) then \(f(R^n)=0\); if \(F(I^n)\subseteq C\) then for all \(\alpha\in R^n\), \(f(\alpha)^{mp}\in C\) or \(R\) embeds in \(M_2(L)\) for \(L\) a field; when \(f=x_1x_2-x_2x_1\) and \(p = 1\) or \(f=x_1x_2+x_2x_1\) and \(p=2\) then \(F(I^2)\subseteq C\) forces \(R\) to be commutative. If instead \(I\) is a nonzero right ideal of \(R\) so \(F(I^n)=0\) then there is \(e^2=e\in\text{soc}(RC)\) so that \(IC=eRC\) and \(f((eRCe)^n)=0\).