【作者】 李庆；

【导师】 王芳贵；

【作者基本信息】 四川师范大学 ， 基础数学， 2010， 博士

【摘要】 论文摘要：本文恒设D是整环,K=qf(D)是D的商域,(?)是D的整闭包,X是D上的一个未定元.在本文的引言中我们概述了本文的研究背景与本文的主要成果.在第一章中我们引入APVMD的概念并主要研究了APVMD的基础理论,比如：APVMD的整闭包,APVMD的扩环,APVMD的多项式环,APVMD与几类重要整环之间的关系,APVMD的经典复合多项式环D+XK[X]等.同时,在这一章中我们继续讨论了AGCD整环的性质.主要证明了整环D是AGCD整环当且仅当D是APVMD且具有挠的t类群.也证明了整环D是AGCD整环当且仅当D是具有挠的t类群的UMT整环且环扩张D(?)(?)是根扩张.本文的第二章继续对第一章所研究的特殊的复合多项式环D+XK[X]进行推广,讨论了更为一般的复合多项式环D+XDS[X],其中S是整环D中的任意浸润乘法集.在讨论这一问题的过程中,我们引入了一种新的研究方法,称为几乎t分裂集,并对几乎t分裂集的基础性质进行了讨论.得出了这一章中的主要结果：复合多项式环D+XDS[X]是APVMD当且仅当D+XDS[X]是well-behaved,D与DS都是APVMD同时S是几乎t分裂集.最后,我们也分别讨论了AP整环和AB整环上的复合多项式环D+XDS[X].证明了若S是D的任意浸润乘法集,则D+XDS[X]是AP整环(AB整环)当且仅当D是AP整环(AB整环)且Ds=K.在第三章我们研究了APVMD上的拉回图的性质,对四种不同类型的拉回图进行了讨论.主要证明了在(△M)型中,R是APVMD当且仅当D和T都是APVMD,TM是AV整环且qf(D)(?)T/M是根扩张.也证明了在(△M*)型中,R是APVMD当且仅当D和T都是APVMD且TM是AV整环.其次,证明了在(△’)型中,假设T是AV整环,则R是APVMD当且仅当D是APVMD且qf(D)(?)T/I是根扩张.最后,主要证明了在(△*)型中,记T=(Iv:Iv),则R是APVMD当且仅当T是APVMD且TI是AV整环,并对D中任意非零素理想(?),或者(1)D(?)和Tφ-1(D\(?))都是AV整环,或者(2)存在D中一有限生成理想A使得A(?)P,A-1∩E=D以及(φ-1(P)T)t=T.根据(t,v)-Dedekind整环是APVMD,因此,在第四章我们主要研究(t,v)-Dedekind整环的局部化问题以及在分次环中的应用.从这些性质当中我们观察到(t,v)-Dedekind整环与APVMD之间的联系与差异.我们主要证明了D是(t,v)-Dedekind整环当且仅当对任意乘法集S(?)Nv,D[X]S是(t,v)-Dedekind整环.我们证明了D是t局部(t,v)-Dedekind整环当且仅当D[X]是t局部(t,v)-Dedekind整环,当且仅当D[X]Nv是t局部(t,v)-Dedekind整环,当且仅当D[X]Nv是局部(t,v)-Dedekind整环.我们也证明了若R=(?)α∈ΓRα是分次整环,则R是(t,v)-Dedekind整环当且仅当R是分次(t,v)-Dedekind整环.利用这些性质,我们最后讨论了(t,v)-Dedekind整环上的群环和半群环.证明了若R是整环,则群环R[X;G]是(t,v)-Dedekind整环当且仅当R是(t,v)-Dedekind整环且G具有型(0,0,…).也证明了半群环R[Γ]是(t,v)-Dedekind整环当且仅当R是(t,v)-Dedekind整环且Γ是(t,v)-Dedekind半群.上面各章的讨论都是在特定的星算子(比如v-算子,t-算子,w-算子)下讨论整环.在第五章我们就把特定的星算子的研究推广到一般星算子上,并引入了AP*MD和A*GCD整环的概念.得出了整环D是AP*MD当且仅当D是APVMD且*s=t.主要证明了D是AP*MD且Cl*s(D)是挠的当且仅当D是A*GCD整环.同时我们也得出了一些推广性的结果,并为APVMD的继续发展提供了新的研究思路.第ⅲ页,共58页更多还原

【Abstract】 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.更多还原