Nilpotent Structures Of Generalized Semicommutative Rings

In this paper,we mainly study the nilpotent structures of generalized semi-commutative rings,especially we study the generalized semi-commutative properties of nilpotent elements on reversible rings,semi-commutative rings and?-semi-commutative rings.A proper subclass of CNZ rings is defined,and we show that the class of one-sided nilpotent semi-commutative rings is strictly placed between the class of semi-commutative rings and that of nil-semi-commutative rings.We also show by a counter-example that an ?-nilpotent semi-commutative ring need not be?-semi-commutative.This paper consists of four chapters.In Chapter 1,some backgrounds and main results of this paper are given.In Chapter 2,we define and study the concepts of left(or right)nilpotent reversible rings.It is proved that the class of left nilpotent reversible rings and the class of right nilpotent reversible rings are proper subclasses of the class of CNZ rings.Various extensions of the left nilpotent reversible rings are described.It is proved that a ring R is a reversible ring if and only if it is a semiprime left nilpotent reversible ring.For the right Ore ring R,we show that if R is a left nilpotent reversible ring,then its classical right quotient ring Q is also left nilpotent reversible.Chapter 3 is a study of the one-sided nilpotent structures of semi-commutative rings.The concepts of left and right nilpotent semi-commutative rings are defined and studied,and we show that this class of rings is a proper generalization of that of semi-commutative rings.Moreover,we also show that the class of one-sided nilpotent semi-commutative rings is strictly placed between the class of semi-commutative rings and that of nil-semi-commutative rings.Some characterizations of the properties of one-sided nilpotent semi-commutative ring are given.As an application,some new equivalent characterizations of semi-prime rings are given.In Chapter 4,we study the nilpotent structure of ?-semi-commutative rings.We show that in general an ?-nilpotent semi-commutative ring is not necessarily an?-semi-commutative ring,and a counter-examples is also given to show that the class of ?-nilpotent semi-commutative rings and the class of semi-commutative rings are two different types of rings.Various properties of ?-nilpotent semi-commutative rings are studied and characterized.It is proved that the polynomial ring R[x] over an?-nilpotent semi-commutative Armendariz ring R is ?-nilpotent semi-commutative.Some well-known results on semi-commutative rings are generalized.