%0 Journal Article
%T On Generalized Derivations in Nearrings
%A Mohd Rais Khan
%A Mohd Mueenul Hasnain
%J Chinese Journal of Mathematics
%D 2013
%R 10.1155/2013/752928
%X 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¨C18] 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£żall . 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£żall . 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£żall . 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¨C26]. In this section, we investigate some results of nearrings satisfying certain identities involving
%U http://www.hindawi.com/journals/cjm/2013/752928/