Derivations cocentralizing polynomials (Q1282136)

The authors prove the following theorem: Let \(R\) be a prime ring with extended centroid \(C\), let \(D\) and \(E\in\text{Der}(R)\), and let \(f(X_1,\dots,X_t)\in C\{X_1,\dots,X_t\}\) and not central valued on \(R\). If for all \(r_i\in R\), \(D(f(r_i))f(r_i)-f(r_i)E(f(r_i))\in C\) then either \(D=E=0\), or \(E=-D\) and \(f(X_i)^2\) is central valued on \(R\), except when \(\text{char }R=2\) and \(\dim_C RC=4\). This result generalizes the corresponding one for a multilinear polynomial \(f(X_j)\) proved by \textit{T.-L. Wong} [Taiwanese J. Math. 1, No. 1, 31-37 (1997; Zbl 0885.16022)].