Some Studies Of GWCN Rings

In this thesis, we study some properties of GWCN rings, and promote GWCN rings. According to the research on GWCN rings, we not only find out the relationship between GWCN rings and other special rings, but also study the regularity and the clean properties of GWCN rings, and discuss their extensions.Firstly, we introduce the relationship between GWCN rings and other special rings, and give some examples to explain that { CN rings } (?){ GWCN rings }(?){ weakly semicommutative rings }, and obtain the conditions when GWCN rings become reduced rings. Then we discuss some extensions of GWCN rings such as matrix extension and localization.Secondly, we study the regularity of GWCN rings, and mainly get some results as follows: (1) a ring R is reduced(?) R is an n-regular, CN ring (?) R is an n-regular,GWCN ring; (2) Let R be a GWCN ring, then R is left weakly regular if and only if R is weakly regular and biregular; (3) if R has an Abelian maximal left ideal and R is a GWCN ring, then R is strongly regular if and only if R is a left GP-V '-ring whose maximal essential left ideals are GW-ideals if and only if R is a left GP-V'-ring whose maximal essential right ideals are GW-ideals. We also discuss the clean properties of GWCN rings, and get the result: R is weakly exchange if and only if R is weakly clean.Finally, we promote GWCN rings and introduce the concept of the ?-GWCN rings. We investigate the relationships between ?-GWCN rings and other special rings and give some properties of a-GWCN rings. We mainly obtain some results as follows: let a be an endomorphism, I is an ideal of a ring R, ?(I)(?) I , then:(1) if I(?) N(R) , R is ?-GWCN, then R/I is ?-GWCN; (2) if I is reduced, R/I is?-GWCN, then R is ?-GWCN. Which a: R/I? R/I, ?(a + I) = ?(a) + I, for all a? R.