Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

Let be a 2-torsion free ring and let be a noncentral Lie ideal of , and let and be two generalized derivations of . We will analyse the structure of in the following cases: (a) is prime and for all and fixed positive integers ; (b) is prime and for all and fixed integers ; (c) is semiprime and for all and fixed integer ; and (d) is semiprime and for all and fixed integer . 1. Introduction Let be an associative ring with characteristic different from 2, its center, its (right) Utumi quotient ring, and its extended centroid. The simple commutator will be denoted by . Recall that a derivation is an additive map satisfying the product rule for all . A left multiplier on a ring is an additive map satisfying the rule for all . In case there exists an endomorphism of such that for all , then is called left -multiplier of . A generalized derivation on a ring is an additive map satisfying for all and some derivation of . A significative example is a map of the form , for some ; such generalized derivations are called inner. Generalized derivations have been primarily studied on operator algebras. Therefore any investigation from the algebraic point of view might be interesting (see, e.g., [1]). Notice that any derivation is a generalized one and also that the generalized inner derivations include left multipliers and right multipliers. Thus the concept of generalized derivation covers both the concept of derivation and the concept of left (right) multipliers. Since the sum of two generalized derivations is a generalized derivation, of course every map of the form is a generalized derivation on , where is a fixed element of and is a derivation of . In [1, Theorem 3] Lee proved that every generalized derivation on a dense right ideal of can be uniquely extended to the Utumi quotient ring of , and thus any generalized derivation of can be defined on the whole ; moreover it is of the form for some and is a derivation on ( is said to be a generalized derivation associated with derivation ). Many results in the literature indicate that the global structure of a ring is often tightly connected to the behaviour of additive mappings defined on . In [2] Bergen proved that if is an automorphism of such that , for all , where is a fixed integer, then . Daif and Bell [3] showed some results which have the same flavour, when the automorphism is replaced by a nonzero derivation . In [3] it is proved that if is a semiprime ring with a nonzero ideal such that , or , for all , then is central. Later Hongan [4] proved that if is a 2-torsion free semiprime ring and a nonzero ideal