Let be a semiprime ring, a nonzero ideal of , and , two epimorphisms of . An additive mapping is generalized -derivation on if there exists a -derivation such that holds for all . In this paper, it is shown that if , then contains a nonzero central ideal of , if one of the following holds: (i) ; (ii) ; (iii) ; (iv) ; (v) for all . 1. Introduction Throughout the present paper, always denotes an associative semiprime ring with center . For any , the commutator and anticommutator of and are denoted by and and are defined by and , respectively. Recall that a ring is said to be prime, if for , implies either or and is said to be semiprime if for , implies . An additive mapping is said to be derivation if holds for all . The notion of derivation is extended to generalized derivation. The generalized derivation means an additive mapping associated with a derivation such that holds for all . Then every derivation is a generalized derivation, but the converse is not true in general. A number of authors have studied the commutativity theorems in prime and semiprime rings admitting derivation and generalized derivation (see e.g., [1–8]; where further references can be found). Let and be two endomorphisms of . For any , set and . An additive mapping is called a -derivation if holds for all . By this definition, every -derivation is a derivation, where means the identity map of . In the same manner the concept of generalized derivation is also extended to generalized -derivation as follows. An additive map is called a generalized -derivation if there exists a -derivation such that holds for all . Of course every generalized -derivation is a generalized derivation of , where denotes the identity map of . There is also ongoing interest to study the commutativity in prime and semiprime rings with -derivations or generalized -derivations (see [9–17]). The present paper is motivated by the results of [17]. In [17], Rehman et al. have discussed the commutativity of a prime ring on generalized -derivation, where and are automorphisms of . More precisely, they studied the following situations: (i) ; (ii) ; (iii) ; (iv) ; (v) for all , where is a nonzero ideal of . The main objective of the present paper is to extend above results for generalized -derivations in semiprime ring , where and are considered as epimorphisms of . To prove our theorems, we will frequently use the following basic identities: 2. Main Results Theorem 2.1. Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a -derivation of such that . If
References
[1]
M. Ashraf, N. Rehman, S. Ali, and M. R. Mozumder, “On semiprime rings with generalized derivations,” Boletim da Sociedade Paranaense de Matemática. 3rd Série, vol. 28, no. 2, pp. 25–32, 2010.
[2]
M. Ashraf, A. Ali, and S. Ali, “Some commutativity theorems for rings with generalized derivations,” Southeast Asian Bulletin of Mathematics, vol. 31, no. 3, pp. 415–421, 2007.
[3]
H. E. Bell and N.-U. Rehman, “Generalized derivations with commutativity and anti-commutativity conditions,” Mathematical Journal of Okayama University, vol. 49, pp. 139–147, 2007.
[4]
Q. Deng and H. E. Bell, “On derivations and commutativity in semiprime rings,” Communications in Algebra, vol. 23, no. 10, pp. 3705–3713, 1995.
[5]
B. Dhara, “Remarks on generalized derivations in prime and semiprime rings,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 646587, 6 pages, 2010.
[6]
B. Hvala, “Generalized derivations in rings,” Communications in Algebra, vol. 26, no. 4, pp. 1147–1166, 1998.
[7]
E. C. Posner, “Derivations in prime rings,” Proceedings of the American Mathematical Society, vol. 8, pp. 1093–1100, 1957.
[8]
M. A. Quadri, M. S. Khan, and N. Rehman, “Generalized derivations and commutativity of prime rings,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 9, pp. 1393–1396, 2003.
[9]
F. Ali and M. A. Chaudhry, “On generalized -derivations of semiprime rings,” Turkish Journal of Mathematics, vol. 35, no. 3, pp. 399–404, 2011.
[10]
N. Arga？ and E. Albas, “On generalized -derivations,” Sibirski？ Matematicheski？ Zhurnal, vol. 43, no. 6, pp. 977–984, 2002.
[11]
N. Arga？, A. Kaya, and A. Kisir, “ -derivations in prime rings,” Mathematical Journal of Okayama University, vol. 29, pp. 173–177, 1987.
[12]
N. Ayd？n and K. Kaya, “Some generalizations in prime rings with -derivation,” Turkish Journal of Mathematics, vol. 16, no. 3, pp. 169–176, 1992.
[13]
？. G？lba？i and E. Ko？, “Some commutativity theorems of prime rings with generalized -derivation,” Communications of the Korean Mathematical Society, vol. 26, no. 3, pp. 445–454, 2011.
[14]
E. G？lba？i and E. Ko？, “Notes on generalized -derivation,” Rendiconti del Seminario Matematico della Università di Padova, vol. 123, pp. 131–139, 2010.
[15]
Y.-S. Jung and K.-H. Park, “On generalized -derivations and commutativity in prime rings,” Bulletin of the Korean Mathematical Society, vol. 43, no. 1, pp. 101–106, 2006.
[16]
H. Marubayashi, M. Ashraf, N. Rehman, and S. Ali, “On generalized -derivations in prime rings,” Algebra Colloquium, vol. 17, no. 1, pp. 865–874, 2010.
[17]
N. U. Rehman, R. M. AL-Omary, and C. Haetinger, “On Lie structure of prime rings with generalized -derivations,” Boletim da Sociedade Paranaense de Matemática, vol. 27, no. 2, pp. 43–52, 2009.
[18]
M. N. Daif and H. E. Bell, “Remarks on derivations on semiprime rings,” International Journal of Mathematics and Mathematical Sciences, vol. 15, no. 1, pp. 205–206, 1992.
