This paper extends Ashraf’s Theorem. We prove that if R is a 2-torsion-free ring which has a commutator right nonzero divisor, then every Jordan generalized higher derivation on R is a generalized higher derivation. | Turk J Math 29 (2005) , 1 – 10. ¨ ITAK ˙ c TUB On Jordan Generalized Higher Derivations in Rings Wagner Cortes∗, Claus Haetinger† Abstract I. N. Herstein proved that any Jordan derivation on a prime ring of characteristic not 2 is a derivation. M. Bre˘sar extended this result to semiprime rings, while M. Ferrero and C. Haetinger extended the result to Jordan higher derivations. Recently, M. Ashraf and N. Rehman considered the question of Herstein for a Jordan generalized derivation. This paper extends Ashraf’s Theorem. We prove that if R is a 2-torsion-free ring which has a commutator right nonzero divisor, then every Jordan generalized higher derivation on R is a generalized higher derivation. Key words and phrases: Higher Derivations, Generalized Higher Derivations, Commutator. Introduction Let R be an associative ring not necessarily with an identity element. A derivation (resp. Jordan derivation) d of R is an additive mapping d: R → R such that d(ab) = d(a)b + ad(b), for every a, b ∈ R (resp. d(a2 ) = d(a)a + ad(a), for every a ∈ R). As it is well-known, every derivation is a Jordan derivation and the converse is, in general, not true. If R is a 2-torsion-free semiprime ring, then by the results of I. N. Herstein and M. 2000 Mathematics Subject Classification: 16W25, 16N60, 16U80. a fellowship granted by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico (CNPq, Brazil). † Partially supported by Funda¸ ca ˜o de Amparo a ` Pesquisa do Estado do Rio Grande do Sul (Fapergs, Brazil) ∗ Supported by 1 CORTES, HAETINGER Bre˘sar, every Jordan derivation of R is a derivation ([3], [4], [8]). It turns out that every Jordan derivation of a 2-torsion-free ring is a Jordan triple derivation ([9], Lemma ). We recall that an additive mapping d : R → R is said to be a Jordan triple derivation if d(aba) = d(a)ba + ad(b)a + abd(a), for every a, b ∈ R. Also, R. Awtar extended the Herstein’s Theorem to Lie ideals ([2], Theorem) by proving that if U is a Lie
