Let \(R\) be an associative ring. An additive mapping \(T\colon R\to R\) is called a centralizer if it is both a left and a right centralizer, i.e. if \(T(xy)=T(x)y\) and \(T(xy)=xT(y)\) for all \(x,y\in R\). If \(R\) is an arbitrary ring and \(T\) is a centralizer, then \(T(xyx)=xT(y)x\) for all \(x,y\in R\). The main result of the paper states that the converse holds whenever \(R\) is a 2-torsion free semiprime ring.