An equation related to two-sided centralizers in prime rings. (Q633915)

For an associative ring \(R\), an additive mapping \(T\colon R\to R\) is called a left (resp. right) centralizer if \(T(xy)=T(x)y\) (resp. \(T(xy)=xT(y)\)) for all \(x,y\in R\). An additive mapping \(T\) is said to be a two-sided centralizer in case \(T\) is a left and right centralizer. The main purpose of this article is to prove the following result. Let \(m\) and \(n\) be positive integers, and \(R\) be a prime ring with \(\text{char}(R)=0\) or \(m+n+1\leq\text{char}(R)\). If \(T\colon R\to R\) is an additive mapping satisfying the condition \(T(x^{m+n+1})=x^mT(x)x^n\) for all \(x\in R\), then \(T\) is a two-sided centralizer.