Let R be a semiprime 2-torsion free ring, and let τ be an endomorphism of R. Under some conditions we prove that a left Jordan τ -centralizer of R is a left τ -centralizer of R. Under the same conditions we also prove that a Jordan τ -centralizer of R is a τ -centralizer of R. We thus generalize Zalar's results to the case of τ -centralizers of R.

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Let R be a prime ring. By a generalized derivation we mean an additive mapping g W R ! R such that g.xy/ D g.x/y C xd.y/ for all x; y 2 R where d is a derivation of R. In the present paper our main goal is to generalize some results concerning derivations of prime rings to generalized derivations of prime rings.

Let R be a prime ring with char R ≠ 2 and let d be a generalized derivation on R . We study the generalized derivation d satisfying any of the following identities: