Some Theorems for Sigma Prime Rings with Differential Identities on Sigma Ideals

There has been considerable interest in the connection between the structure and the -structure of a ring, where denotes an involution on a ring. In this context, Oukhtite and Salhi (2006) introduced a new class or we can say an extension of prime rings in the form of -prime ring and proved several well-known theorems of prime rings for -prime rings. A continuous approach in the direction of -prime rings is still on. In this paper, we establish some results for -prime rings satisfying certain identities involving generalized derivations on -ideals. Finally, we give an example showing that the restrictions imposed on the hypothesis of the various theorems were not superfluous. 1. Introduction Throughout the paper, will denote an associative ring with center . For any , the symbol stands for the Lie product and the symbol denotes the Jordan product . A ring is called 2-torsion free, if whenever , with , then . Recall that a ring is prime if, for any implies ？？or . A ring equipped with an involution is to be -prime if ？？or . An example, according to Oukhtite and Salhi [1], shows that every prime ring can be injected in -prime ring and from this point of view -prime rings constitute a more general class of prime rings. An ideal is a -ideal if is invariant under ; that is, . Note that an ideal may not be a -ideal. Let be a ring of integers and . Consider a map defined by for all . For an ideal of , is not a -ideal of since . Several authors have studied the relationship between the commutativity of a ring and the behavior of a special mapping on that ring. In particular, there has been considerable interest in centralizing automorphisms and derivations defined on rings (see, e.g., [2–4], where further references can be found). As defined in [5, 6], an additive mapping is called generalized derivation with associated derivation if Familiar examples of generalized derivations are derivations and generalized inner derivations and later included 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 a fixed element of and , a derivation of , is a generalized derivation, and if has , all generalized derivations have this form. In 2006, Oukhtite and Salhi [1] introduced a new class or we can say an extension of prime rings in the form of -prime ring. However, the actual motivation behind their first successful work came from Posner’s [7] second theorem only. In [8], they successfully extended the result for -prime ring. Recently, a major