The aim of this paper is to investigate some results of nearrings satisfying certain identities involving generalized derivations. Furthermore, we give some examples to demonstrate the restrictions imposed on the hypothesis of various results which are not superfluous. 1. Introduction The study of derivations in rings goes back to 1957 when Posner [1] proved that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. Many results in this vein were obtained by a number of authors [2–18] in several ways. In view of [19], the concept of generalized derivation is introduced by Hvala [20]. Familiar examples of generalized derivations are derivations and generalized inner derivations, and later includes left multiplier, that is, an additive mapping satisfying for？all . Since the sum of two generalized derivations is a generalized derivation, every map of the form , where is fixed element of and a derivation of , is a generalized derivation; and if has , all generalized derivations have this form. Throughout the paper, will denote a zero symmetric right abelian nearring with multiplicative center . For all , as usual and will denote the well-known Lie and Jordan products, respectively. A nonempty subset of will be called a semigroup right ideal (resp., semigroup left ideal) if ( ). Finally, is called a semigroup ideal if it is a right as well as a left semigroup ideal. A nearring is called prime, if or for？all . We refer to Pilz [21] for the basics definitions and properties of nearring. As noted in [22], an additive mapping is called a derivation of if holds for？all . An additive mapping is said to be a right generalized derivation associated with if and is said to be a left generalized derivation associated with if is said to be a generalized derivation associated with if it is a right as well as a left generalized derivation associated with . The purpose of this note is to prove some results which are of independent interest and related to generalized derivations on nearrings. 2. Ideals and Generalized Derivations in Nearrings Over the last several years, many authors [7, 19, 20, 23] studied the commutativity in prime and semiprime rings admitting derivations and generalized derivations. On other hand, there are several results asserting that prime nearrings with certain constrained derivations have ring-like behavior. It is natural to look for comparable results on nearrings, and this has been done [22, 24–26]. In this section, we investigate some results of nearrings satisfying certain identities involving
References
[1]
E. C. Posner, “Derivations in Prime rings,” Proceedings of the American Mathematical Society, vol. 8, pp. 1093–1100, 1957.
[2]
R. Awtar, “Lie and Jordan structure in prime rings with derivations,” Proceedings of the American Mathematical Society, vol. 41, pp. 67–74, 1973.
[3]
H. E. Bell and W. S. Martindale, “Centralizing mapping of semiprime rings,” Canadian Mathematical Bulletin, vol. 30, pp. 92–101, 1987.
[4]
H. E. Bell and W. S. Martindale, “Semiderivations and commutativity in prime rings,” Canadian Mathematical Bulletin, vol. 31, pp. 500–508, 1988.
[5]
M. Bresar and J. Vukman, “On left derivations and related mappings,” Proceedings of the American Mathematical Society, vol. 110, pp. 7–16, 1990.
[6]
L. O. Chung and J. Luh, “On semicommuting automorphisms of rings,” Canadian Mathematical Bulletin, vol. 21, pp. 13–16, 1978.
[7]
M. N. Daif and H. E. Bell, “Remarks on derivation on semi-prime rings,” International Journal of Mathematics and Mathematical Sciences, vol. 15, no. 1, pp. 205–206, 1992.
[8]
B. Felzenswalb and A. Giambruno, “A commutativity theorem for rings with derivations,” Pacific Journal of Mathematics, vol. 102, pp. 41–45, 1982.
[9]
Y. Hirano, A. Kaya, and H. Tominaga, “On a theorem of mayne,” Mathematical Journal of Okayama University, vol. 25, pp. 125–132, 1983.
[10]
M. Hongan and A. Trzepizur, “On generalization of a theorem of posner,” Mathematical Journal of Okayama University, vol. 27, pp. 19–23, 1985.
[11]
A. Kaya, “A theorem on semicentralizing derivations of prime rings,” Mathematical Journal of Okayama University, vol. 27, pp. 11–12, 1985.
[12]
A. Kaya, “Semicentralizing derivations in prime rings,” DOGA, vol. 11, pp. 100–105, 1987.
[13]
A. Kaya and C. Koc, “Semicentralizing automorphisms of prime rings,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 38, no. 1–4, pp. 53–55, 1981.
[14]
C. Lanski, “Differential identities, Lie ideal and Posner's theorems,” Pacific Journal of Mathematics, vol. 134, pp. 275–297, 1988.
[15]
P. H. Lee and T. K. Lee, “Derivations centralizing symmetric or skew elements,” Bulletin of the Institute of Mathematics Academia Sinica, vol. 14, pp. 249–256, 1986.
[16]
T. Mayne, “Centralizing automorphism of prime rings,” Canadian Mathematical Bulletin, vol. 19, pp. 113–115, 1976.
[17]
T. Mayne, “Ideals and centralizing mappings in prime rings,” Proceedings of the American Mathematical Society, vol. 86, pp. 211–212, 1982, Erratum: vol. 89, pp. 187, 1983.
[18]
T. Mayne, “Centralizing mappings of prime rings,” Canadian Mathematical Bulletin, vol. 27, pp. 122–126, 1984.
[19]
M. Bresar, “On the distance of composition two derivation to the generalized derivation,” Glasgow Mathematical Journal, vol. 33, pp. 89–93, 1991.
[20]
B. Hvala, “Generalized derivations in rings,” Communications in Algebra, vol. 26, no. 4, pp. 1147–1166, 1998.
[21]
G. Pilz, Near-Rings, North Holland, Amsterdam, The Netherlands, 2nd edition, 1983.
[22]
X. K. Wang, “Derivations in prime nearrings,” Proceedings of the American Mathematical Society, vol. 121, pp. 361–366, 1994.
[23]
M. A. Quadri, M. Shadab 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.
[24]
H. E. Bell and G. Mason, “On derivations in near-rings,” North-Holland Mathematics Studies, vol. 137, pp. 31–35, 1987.
[25]
H. E. Bell, On Derivations in Near-Rings II, Kluwer Academic, Dodrecht, The Netherlands, 1997.
[26]
H. E. Bell, “Derivations, products of derivations, and commutativity in near-rings,” Algebra Colloquium, vol. 8, no. 4, pp. 399–407, 2001.
