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Keywords:
semiprime ring; left centralizer; centralizer; Jordan centralizer
semiprime ring; left centralizer; centralizer; Jordan centralizer
Summary:
Let $\Cal K$ be a semiprime ring and $T:\Cal K\rightarrow \Cal K$ an additive mapping such that $T(x^2)=T(x)x$ holds for all $x\in \Cal K$. Then $T$ is a left centralizer of $\Cal K$. It is also proved that Jordan centralizers and centralizers of $\Cal K$ coincide.
References:
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[2] Brešar M., Zalar B.: On the structure of Jordan $\ast $-derivations. Colloquium Math., to appear. MR 1180629
[3] Herstein I.N.: Topics in ring theory. University of Chicago Press, 1969. MR 0271135 | Zbl 0232.16001
[4] Herstein I.N.: Theory of rings. University of Chicago Press, 1961.
[5] Johnson B.E., Sinclair A.M.: Continuity of derivations and a problem of Kaplansky. Amer. J. Math. 90 (1968), 1067-1073. MR 0239419 | Zbl 0179.18103
[6] Šemrl P.: Quadratic functionals and Jordan $\ast $-derivations. Studia Math. 97 (1991), 157-165. MR 1100685
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