Flat Well-centered Overrings of An Integral Domain

Flat Well-centered Overrings of An Integral Domain

蔡金萍

指導教授 ： 劉容真

國立臺灣師範大學/理學院/數學系/碩士(2003年)

摘要

Let A be an integral domain with field of fractions K.
We investigate the structure of the overrings B of A that are
well-centered on A. We consider the relation of well-centeredness to the properties of flatness, localization and sublocalization for B over A.

關鍵字

flat ； overrings ； well-centered ； sublocalization

並列摘要

All rings we consider here are assumed to be commutative with unity. If A is an integral
domain with ‾eld of fractions K, we refer to a subring B of K with A µ B as an overring
of A.
In Section 1, we give an important result that an overring B of A is a °at A-module if and
only if BM = AMA for every maximal ideal M of B. In particular, if B is a °at overring of
A, then B = TAP where P runs over some set of prime ideals of A.
We say that B is well-centered on A if for each b 2 B there exists a unit u 2 B such
that ub = a 2 A. Let S be a multiplicatively closed subset of nonzero elements of A. We
know that the localization AS which is an overring of A is °at over A and well-centered on
A. The converse of this result, however, is not true in general. We observe in Theorem 4.1
that a simple °at overring B generated by a unit of B is a localization of A. Thus we
have in Corollary 4.3 that a simple °at well-centered overrings are localizations. Moreover,
if A is a one-dimensional integral domain, we prove in Theorem 4.20 that every ‾nitely
generated °at well-centered overring of A is a localization of A. On the other hand, we give
in Theorem 3.22 that there exists a Dedekind domain A having a °at well-centered overring
that is not a localization.
The overring B of A is a sublocalization of A if B is an intersection of localizations of A.
In Proposition 2.6, we show that an overring B of A is a sublocalization of A if and only if
B = TfAP : P 2 SpecA and B µ AP g. We give in Theorem 5.4 necessary and su±cient
conditions for each sublocalization overring of a Noetherian domain A to be a localization
of A. If A is a one-dimensional integral domain with Noetherian prime spectrum, we give in
Theorem 5.8 that every sublocalization is °at over A.
This thesis is base on the paper "Well-centered overrings of an integral domain" by William
Heinzer and Moshe Roitman[HM].

並列關鍵字

flat ； overrings ； well-centered ； sublocalization

參考文獻

[AM] M.F. Atyiah and I.G. Macdonald,

Introduction to Commutative Algebra,

[BH] J. Brewer and W. Heinzer,

Associated primes of principal ideals,

Duke Math. J. 41(1974), 1-7.