Generalized Derivations of Semiprime Rings : Some Recent Results of Generalized Derivations on Prime and Semiprime Rings

Let R be an associative ring with center Z(R) . One of the important problems of Ring Theory is to determine whether this ring is commutative or not. The main purpose of this book to shows that the structure of a semiprime ring is very tightly determined by the imposition of a special behavior on one of these derivations(resp.generalized derivation, -derivation and Bi-derivation). We have studied some of these results and supply the details of the proofs and we also prove the results that appeared without proof, we illustrate some of these results by examples. Moreover, we add some results that seems to be new to the best of our knowledge. As usual, [x,y] will denote the commutator and (xoy) anti-commutator.