The main purpose of this paper is to prove the following result. Let R be a noncommutative prime ring of characteristic different from two and let D and G = 0 be (\alpha, beta)-derivations of R into itself such that G commutes with alpha and beta. If [D{x), G(x)] = 0 holds for all x is an eleemnt of R then D = lambdaG where lambda is an element from the extended centroid of R.
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The main purpose of this paper is to investigate additive mapping D : R -> R, where R is a (m + n +1)! and \m2 + n2 - m - n - 4mn\ -torsion free semiprime ring with the identity element, satisfying the relation 2D(xm+n+l) = (m+-n+1)(xmD(x)xn +-xnD(x)xm), for all is an element of R and some integers m > 1, n > 1, m2 + n2 - m - n - 4mn /=0.
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The purpose of this paper is to prove the following result: Let R be a (m+n + 2)! and 3m2n + 3mn2 + 4m2 + 4n2 +10mn-torsion free semiprime ring with an identity element and let T : R -R be an additive mapping such that 3T(xm+n+1) = T(x)xm+n + xmT(x)xn + xm=nT(x) is fulfilled for all x is an element R and some fixed nonnegative integers m and n, m+n=0. In this case T is a centralizer.
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In this paper we investigate identities with alfa-derivations on prime and semiprime rings. We prove, for example, the following result. If D : R - R is an alfa-derivation of a 2 and 3-torsion free semiprime ring R such that [D(x},x2] = 0 holds, for all x is an element of R, then D maps R into its center. The results of this paper are motivated by the work of Thaheem and Samman [20].
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