Properties Of Derivations In Prime Ring And ?-Prime Ring

Theory of Rings is the base of Algebraic Geometry and Algebraic Number Theory.It has rich content and profound background.With the development of other branches of Mathematics,The research of theory of rings is endowed with new content.Derivations in ring is a kind of generalization of differential in Algebra.It is a hot point in mathematics.Especially the properties of derivations act as homomorphisms or anti-homomorphisms on ideals of prime ring are the hot points of researchers.The study of this subject greatly enriched the content of the Classical Theory of Rings.This thesis deals with some properties of derivations act as homomorphisms or anti-homomorphisms on ideals of prime ring.We mainly discusses properties of generalized derivation,(?,?)-derivation,generalized(?,?)-derivation,left derivation,generalized left derivation,left(?,?)-derivation acts as homomorphism or anti-homomorphism on nonzero ideal,Lie ideal and Jordan ideal of prime ring respectively.On the other hand,We expand the research field.We generalize results in prime ring to ?-prime ring and get some new results.It plays an important role to depict the structure of rings.The thesis consists of three chapters.The main contents are as follows:The first chapter mainly introduces the background of this thesis.We analysis the progressof researching of domestic and foreign scholars and obtain some important results.At the same time,we introduce basic concepts and some commonly used symbols in this thesis.The second chapter studies properties of several types of derivations act as homomorphi sms or anti-homomorphisms on nonzero ideals of prime ring.This work is to explore the properties of derivations act as homomorphisms or anti-homomorphisms on Lie ideals of prime ring.We prove the properties of left derivations act as homomorphisms or anti-homomorphisms on Lie ideals of prime ring with the results obtained before.The third chapter expand the field of researching.We generalize some results in prime ring to new results in ?-prime ring.We study some properties of derivations act as homomorphisms or anti-homomorphisms on ?-Lie ideals and ?-Jordan ideals of ?-prime ring and give some commuting conditions of derivations and ? with some properties obtained before.Property of derivations act as homomorphisms or anti-homomorphisms is one of the most important properties of derivations.This results we study in the thesis complement and improve the theory of derivations.