Left centralizers on rings that are not semiprime. (Q640268)

In any ring \(R\), an additive \(T\colon R\to R\) is a (left) centralizer on \(R\) if \(T(xy)=T(x)y\) for all \(x,y\in R\), and is a Jordan centralizer when \(T(xy+yx)=T(x)y+T(y)x\). The main result of the paper is that for any Jordan centralizer \(T\) of \(R\), if \(I\) is the \(T\)-invariant ideal of \(R\) generated by \(\{T(xy)-T(x)y\mid x,y\in R\}\), then \(I\) is a union of nilpotent ideals. Thus \(T\) is a centralizer on \(R/I\). This extend a result of \textit{B. Zalar} [Commentat. Math. Univ. Carol. 32, No. 4, 609-614 (1991; Zbl 0746.16011)] that a Jordan centralizer is a centralizer when \(R\) is semiprime and 2-torsion free.