Derivations And Generalized Derivations In Prime Rings

Since 1957 when E. C. Posner initiated the famous Posner theorem, the research of centralizing mappings, commuting mappings and derivations in prime or semiprime rings has become an important field in the theory of ring, especially over the last twenty years many authors have studied Posner's theorem, in more generalized versions. They showed that, the structure of a ring is very tightly determined by the imposition of a special behavior on one of its derivations. The goal of this work is to generalize some of these results and present others, which can be considered as a contribution to the theory of derivations in prime rings.(1) In Chapter 1, as preliminaries, we will first review some basic definitions and preliminary results. And then, several lemmas are proved.(2) In Chapter 2, we will discuss the action of derivations on Lie ideals in prime rings, and many conditions which force the Lie ideals to be central are given. Moreover, we will also give some results about a pair of derivations on Lie ideals.(3) In Chapter 3, we will discuss the relation between the action of generalized derivations and the structure of prime rings.(4) In Chapter 4, some other results about semi-derivations in prime rings will be introduced.