We consider Jordan derivations of a unital algebra A having a nontrivial idempotent. It turns out that on unital algebras there exist Jordan derivations that are not derivations. For this purpose we introduce the term a singular Jordan derivation, which is a proper Jordan derivation of the form that depends on Peirce decomposition of the unital algebra A. Singular Jordan derivations are usually antiderivations. The main result of the paper states that under certain conditions every Jordan derivation of A is the sum of a derivation and a singular Jordan derivation.

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Let n be a fixed integer, let R be an (n+1)!-torsion free semiprime ring with the identity element and let F: R → R be an additive mapping satisfying the relation [Formula: see text] for all x ∈ R. In this case, we prove that F is of the form 2F(x)=D(x)+ax+xa for all x ∈ R, where D: R → R is a derivation and a ∈ R is some fixed element.

Abstract. In this paper functional equations related to derivations on semiprime rings and standard operator algebras are investigated. We prove, for example, the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X into itself and let A(X) ⊂ L(X) be a standard operator algebra. Suppose there exist linear mappingsThroughout, R will represent an associative ring with center Z(R). As usual we write [x, y] for xy − yx. Given an integer n ≥ 2, a ring R is said to be n−torsion free, if for x ∈ R, nx = 0 implies x = 0. Recall that a ring R is prime, if for a, b ∈ R, aRb = (0) implies a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. We denote by Q s the symmetric Martindale ring of quotients. For the explanation of Q s we refer the reader to [2]. Let A be an algebra over the real or complex field and let B be a subalgebra of A.

The purpose of this paper is to prove the following result. Let n≥3 be some fixed integer and let R be a (n+1)!2n-2-torsion free semiprime unital ring. Suppose there exists an additive mapping D: R→ R satisfying the relation for all x ∈ R. In this case D is a derivation. The history of this result goes back to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion free prime ring is a derivation.