The Structure And Classification Of An APVMD And Its Applications

Let D be an integral domain with quotient field K=qf(D), D an integral closure of D, and X an indeterminate over D. In the introduction, we introduce the background and main results of this thesis. In chapter 1, we introduce the notion of an APVMD and study the basic results, such as, the integral closure, the overring, the polynomial ring and the connection between an APVMD and several important domains, and the special composite poly-nomial ring D+XK[X]. Also, in this chapter we continue to investigate the study of AGCD-domains. We mainly prove that an integral domain D is an AGCD-domain if and only if D is an APVMD with torsion t-class group. Also, we show that D is an AGCD-domain if and only if D is a UMT-domain with torsion t-class group and the extension D C D is a root extension.In chapter 2, we will generalize the special composite polynomial ring D+ XK[X] to more general composite polynomial ring D+XDS[X], where S is a saturated multiplicatively closed set of D. During the discussion, we introduce a notion of an almost t-splitting set to develop the polynomial ring D+XDS[X]. We investigate some basic properties of an almost t-splitting set. And we get the main result of this chapter that a polynomial composite ring D+XDS[X] is an APVMD if and only if D+XDS[X] is well-behaved, both D and DS are APVMDs, and S is an almost t-splitting set. Also, we discuss the composite polynomial ring D+XDS[X] of an AP-domain and an AB-domain respectively. We prove that if S is a saturated multiplicatively closed set of D, then D+ XDS[X] is an AP-domain (AB-domain) if and only if D is an AP-domain (AB-domain) and DS= K.In chapter 3, we mainly investigate four types of pullbacks over APVMDs. The purpose of chapter 3 is to deal with the transfer of the notion "APVMD" to the pullbacks of different types and to continue the study of AP-domains and AV-domains in the pullbacks. We prove that for the diagram (ΔM), R is an APVMD if and only if D and T are APVMDs, TM is an AV-domain and the extension qf(D)(?)T/M is a root extension. Also, we prove that for the diagram (ΔM*), R is an APVMD if and only if D and T are APVMDs and TM is an AV-domain. And we show that for the diagram (Δ'), assume that T is an AV-domain, then R is an APVMD if and only if D is an APVMD and the extension qf(D)(?)T/I is a root extension. Also, we indicate that for a diagram (Δ*), assume that T= (Iv:Iv), then R is an APVMD if and only if T is an APVMD and TI is an AV-domain, and for each nonzero prime ideal P of D, either (1) Dp and Tφ-1(D\P) are AV-domains, or (2) there is a finitely generated ideal A of D such that A(?)P, A-1∩E=D, and (φ-1(P)T)t=T.Since a (t,v)-Dedekind domain is an APVMD, in chapter 4 we mainly continue to investigate an (t,v)-Dedekind domain, such as, the localization, the application in graded rings and so on. Using these properties, we can see the difference between an (t,v)-Dedekind domain and an APVMD. We mainly prove that an integral domain D is an (t,v)-Dedekind domain if and only if for each multiplicative set S(?)Nv, D[X]S is an (t,v)-Dedekind domain. We also show that an integral domain D is t-locally a (t,v)-Dedekind domain, if and only if D[X] is t-locally a (t,v)-Dedekind domain, if and only if D[X]Nv is t-locally a (t,v)-Dedekind domain, and if and only if D[X]Nv is locally a (t,v)-Dedekind domain. Also, we show that if R=(?)α∈ΓRαis a graded domain, then R is a (t,v)-Dedekind domain if and only if R is a graded (t,v)-Dedekind domain. As an application, we discuss the group ring and the semigroup ring over a (t,v)-Dedekind domain. We prove that the group ring R[X; G] is a (t,v)-Dedekind domain if and only if R is a (t,v)-Dedekind domain and G has type (0,0,…). We show that the semigroup ring R[Γ] is a (t,v)-Dedekind domain if and only if R is a (t,v)-Dedekind domain andΓis a (t,v)-Dedekind semigroup.In the final chapter, we generalize the special star operations, such as v-operation,t-operation and w-operation, to more general star operations. Also, we extend the notions of APVMD and AGCD to the notions of AP*MD and A*GCD respectively. It is easy to know that an integral domain D is an AP*MD if and only if D is an APVMD and*s=t. We prove that D is an A*GCD-domain if and only if D is an AP*MD with torsion*s-class group of D.Also, We get some general results, which provide us with a new approach to an APVMD.