Properties Of Generalized(?,?)-derivations In Prime Rings And Antiderivations In ?-Prime Rings

Ring theory is an important branch of algebra.It is the basis of algebraic geometry and algebraic number theory.Many new concepts and methods are introduced into ring theory with the development of other branches of mathematics.Derivation belongs to the field of operator algebra.It is an improvement of differential in algebra.Posner studied the structure of rings by using the properties of derivations in the ring in 1957.He proposed the famous Theorem.Since then,derivations in rings have become a hot topic in algebra.Many scholars have improved Posner Theorem from different angles in various ways.A main mode of improving Posner Theorem is to change the certain algebraic conditions that derivations satisfy on special subsets of rings.The special subsets considered are often the ideals,Lie ideals and Jordan ideals of rings.Many generalized and derived derivations such as generalized(?,?)-derivations and antiderivations emerged with the development of derivation theory.It is very meaningful to improve properties of derivations in prime rings to these generalized and derived derivations.Oukhtite and Salhi proposed a more general concept than prime ring called ?-prime ring in 2006.Many scholars improved current achieve-ments in prime rings to obtain corresponding results in ?-prime rings and obtained a series of valuable results.We research properties of generalized(?,?)-derivations on nonzero ideals and Lie ideals of prime rings on the basis of current achievements about derivations.Furthermore,we research properties of antiderivations on nonzero ?-Lie ideals and?-Jordan ideals of ?-prime rings by means of algebraic methods such as substit-ution and linearization in this thesis.We do new work from 3 aspects:we improve properties of derivations in prime rings or in ?-prime rings to generalized(?,?)-derivations and antiderivations;we change the certain algebraic conditions that derivations satisfy in rings or on special subsets of rings.We mainly discuss properties of derivations on special subsets of rings as homomorphisms and anti-homomorphisms.And then,we will describe the the structure of rings with our research achievements;We improve properties of derivations in prime rings to corresponding results in ?-prime rings.At the same time,we change the special subsets of rings on which the certain algebraic conditions hold that derivations satisfy.We generalize Lie ideals and Jordan ideals to ?-Lie ideals and ?-Jordan ideals respectively.This thesis is made up of four parts:We describe the research background and significance.Then,we analyse related research at home and abroad and introduce basic concepts and identities in the thesis.Furthermore,we briefly display the research content and innovations of this thesis in Chapter 1.We study some properties of generalized(?,?)-derivations on nonzero ideals and nonzero square closed Lie ideals of prime rings in Chapter 2.We research some properties of reverse derivations on non-zero ?-Lie ideals and non-zero ?-Jordan ideals of ?-prime rings in Chapter 3.Finally,we summarize the thesis in Chapter 4.