Siberian Mathematical Journal, Vol. 56, No. 2, pp. 199–205, 2015 Original Russian Text Copyright c  2015 Aboubakr A. and Gonz´ alez S. GENERALIZED REVERSE DERIVATIONS ON SEMIPRIME RINGS A. Aboubakr and S. Gonz´ alez UDC 512.552.34 Abstract: We generalize the notion of reverse derivation by introducing generalized reverse derivations. We define an l-generalized reverse derivation (r-generalized reverse derivation) as an additive mapping F : R → R, satisfying F (xy)= F (y)x + yd(x)(F (xy)= d(y)x + yF (x)) for all x, y ∈ R, where d is a reverse derivation of R. We study the relationship between generalized reverse derivations and generalized derivations on an ideal in a semiprime ring. We prove that if F is an l-generalized reverse (or r-generalized) derivation on a semiprime ring R, then R has a nonzero central ideal. DOI: 10.1134/S0037446615020019 Keywords: semiprime ring, ideal, derivation, reverse derivation, l-generalized derivation, r-generalized derivation, l-generalized reverse derivation, r-generalized reverse derivation 1. Introduction Throughout this paper R denotes an associative ring with center Z (R). If I is a subset of R, then C R (I ) denotes the centralizer of I which is defined by C R (I )= {x ∈ R | xa = ax for all a ∈ I }. Recall that R is prime if aRb = (0) implies that a = 0 or b = 0. The ring R is semiprime if aRa =0 implies a = 0 (obviously, every prime ring is semiprime). As usual, [x, y] denotes the commutator xy - yx. We will make extensive use of the basic commutator identities [xy, z ]= x[y,z ]+[x, z]y and [x, yz ]= y[x, z]+[x, y]z . An additive mapping d from R into itself is called a derivation if d(xy)= d(x)y + xd(y) for all x, y ∈ R. Given a ∈ R, the additive mapping d : R → R defined by d(x)=[x, a] for all x ∈ R is a derivation called the inner derivation of R determined by a. The notion of reverse derivation arose in one early paper of Herstein [1], when he studied Jordan derivations on prime associative rings. The notion of reverse derivation has relations with some general- izations of derivations. A reverse derivation is an additive mapping d from a ring R into itself satisfying d(xy)= d(y)x + yd(x) for all x, y ∈ R. So, each reverse derivation is a Jordan derivation (but the converse is not true in general). In the anticommutative case each reverse derivation is an antiderivation, and each antiderivation is a reverse derivation. The reverse derivations in the case of prime Lie and prime Malcev algebras were studied by Hopkins and Filippov. Those papers provided some examples of nonzero reverse derivations for the simple 3-dimensional Lie algebra sl 2 (see [2]) and characterized the prime Lie algebras admitting a nonzero reverse derivation (see [3, 4]). In particular, Filippov proved that each prime Lie algebra, admitting nonzero reverse derivation is a PI-algebra. Filippov also described all reverse deriva- tions of prime Malcev algebras [5]. The supercase of reverse derivations (antisuperderivations) of simple Lie superalgebras was studied by Kaygorodov in [6] and [7]. He proved that every reverse superderivation of a simple finite-dimensional Lie superalgebra over an algebraically closed field of characteristic zero is the zero mapping. After that, Kaygorodov proved that every r-generalized reverse (or l-generalized) derivation of a simple (non-Lie) Malcev algebra is the zero mapping (see [8]). The first author was supported by the Erasmus Mundus Programme for the financial support of the PhD MEDASTAR Program (Grant 2011–4051/002–001–EMA2). The second author was partially supported by Project MTM2010–18370–C04–01. Fayoum and Oviedo. Translated from Sibirski˘ ı Matematicheski˘ ı Zhurnal, Vol. 56, No. 2, pp. 241–248, March–April, 2015. Original article submitted February 4, 2014. 0037-4466/15/5602–0199 c  199