The Linked Property Of Rings And The Split Property Of Modules In W-modularization Theory

In this thesis,in Chapter 1,some known related knowledge of w-module theory,graded rings,graded w-modules,w-projective modules and w-split modules are introduced.In Chapter 2,firstly,the concept of w-linked homomorphisms is introduced,namely,let φ:R→T be a ring homomorphism.If T as an R-module is a w-module,then φ is called a w-linked homomorphism.Secondly,the concepts of the wφ-operation and DWφ rings are introduced,and some properties of DWφ rings and the relationships between DWφ rings and DW rings over a w-linked homomorphism φ:R→ T are discussed.Thirdly,with the help of w-linked homomorphisms and DWφ rings,it is shown that if T is not a DW ring,then T must have an infinite number of maximal w-ideals.Then,with the help of w-linked homomorphisms,some properties of w-factor rings are discussed,and some classical results on the quotient rings have corresponding statements on the w-factor rings.For example,it is shown that an integral domain R is an SM-domain with w-dim（R）≤1,if and only if for any nonzero w-ideal I of R,（R/I）w is an Artinian ring,if and only if for any nonzero element a ∈ R,（R/（a））w is an Artinian ring.In this chapter,finally,we give some applications of w-linked homomorphisms,it is shown that the w-factor ring of a w-Noetherian ring is a w-Noetherian ring.And for a special pullback graph（RDTF,M）,it is shown that R is a DW domain,if and only if D is a DW domain and T is a DWφ domain.Let G be a multiplicative Abelian group with identity element e,and（?）Rσ is a commutative G-graded ring with identity 1.In Chapter 3,firstly,the concepts of graded w-envelope of a module and graded w-Noetherian rings are introduced.Then some equivalent characterizations of graded w-envelope of a module and graded w-Noetherian rings are shown.Next,it is shown that the strongly graded ring R is w-linked extension of Re.Meanwhile,it is shown that let R be a strongly graded ring.Then R is a graded w-Noetherian ring if and only if Re is a w-Noetherian ring.Besides,for the general G-graded case,the principal ideal theorem on graded w-Noetherian domains is proved.More precisely,let R be a graded w-Noetherian domain and let a∈R be a homogeneous element.Suppose p is a minimal graded prime ideal of（a）.Then the graded height of a graded prime ideal p is at most 1.Finally,in Chapter 4,the concept of w-split modules is introduced,and some properties of iw-split modules and the relationships between iw-split modules and injective modules are discussed.Then with the help of iw-split modules,new equivalent characterizations of DW rings,semi-simple rings and hereditary rings are given.It is shown that R is a DW ring if and only if every iw-split module is injective;R is a semi-simple ring if and only if every R-module is iw-split;R is a hereditary ring if and only if every factor module of an iw-split module is iwsplit.Besides,the concept of w-projective dimensions is introduced,some related properties of w-projective dimensions are studied.Based on the concept of w-projective dimensions,the concept of the global w-projective dimensions of rings is introduced,it is shown that R is a semi-simple ring if and only if the global w-projective dimension of R is 0.And then we study the ring with the global w-projective dimension at most 1.