On overrings of a domain

Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry Pub Date : 1969-01-01 DOI:10.32917/HMJ/1206138590

H. Butts, N. Vaughan

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引用次数: 4

Abstract

Throughout this paper D will denote an integral domain with 1^0 and quotient field X, and by an overring of D will be meant a ring / such that DczJczK. An ideal A of D is called a valuation ideal provided there exists a valuation overring Dv of D such that ADV nD = A ([22; 340], [10]). If 77 is a general ring property, then we shall refer to an ideal A of D as a Π-ideal provided there exists an overring / of D such that / is a 77-domain (i.e. / has the property 77) and A = AJπD. It is shown in [10] that if every principal ideal of D is a valuation ideal, then D is a valuation ring. Furthermore, if every proper ideal of D is a Dedekind ideal, then D is a Dedekind domain [2] and if every proper ideal of D is a Prufer ideal, then D is a Prufer domain [7], [10; 238]. In this paper we are mainly concerned with the following question. When does the statement (a) "there exists a collection d of 77-ideals of 7)" imply the statement (b) "D is a 77-domain" (i.e. D has property 77)? Our main result in this direction is that (a) implies (b) when "77-domain" = "Krull domain" and d is the collection of proper principal ideals of 7), i.e. if every proper principal ideal is a Krull ideal, then D is a Krull domain. The same result holds in case "Krull domain is replaced by either "integrally closed domain" or "completely integrally closed domain". In addition we show that (a) implies (b) when d is the collection of proper finitely generated ideals of D and 77 is any of the following ring properties: Prufer, 1-dim. Prufer, almost Dedekind, or Dedekind. We remark that (a) does not always imply (b), even in the case that d is the set of all ideals of D (e.g. if 77 is one of P.I.D., Bezout, or (λR-propertysee Section 5). In general we use the notation and terminology of [21] and [22]. In particular, c denotes containment, while < denotes proper containment and A is a proper ideal of D provided (0)

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域的上环

在本文中，D表示一个有1^0和商域X的积分域，D的上环表示一个满足DczJczK的环。如果存在一个覆盖在D的Dv上的估值，使得ADV nD = A ([22;340],[10])。如果77是一般环性质，则我们将D的理想a称为Π-ideal，只要D的上环/使/是77-域(即/具有性质77)且a = AJπD。由[10]可知，若D的每一个主理想都是估值理想，则D是估值环。更进一步，如果D的所有真理想都是Dedekind理想，则D是Dedekind定义域[2];如果D的所有真理想都是Prufer理想，则D是Prufer定义域[7]，[10];238]。在本文中，我们主要关注以下问题。语句(a)何时生效?“存在77个理想的集合”暗示了语句(b)。“D是77域”(即D具有属性77)?在这个方向上我们的主要结果是(a)蕴涵(b)当“77-域”=“Krull域”且d是7)的适当主理想的集合，即如果每个适当主理想都是Krull理想，则d是一个Krull域。同样的结果也适用于“Krull域”被“整闭域”或“完全整闭域”取代的情况。此外，我们证明了(a)蕴涵(b)，当d是d的适当有限生成理想的集合，并且77是下列环性质中的任何一个:Prufer, 1-dim。普鲁弗，差不多是戴德金，或者戴德金。我们注意到(a)并不总是暗示(b)，即使在d是d的所有理想的集合的情况下(例如，如果77是P.I.D, Bezout或(λ r -)性质之一，参见第5节)。通常我们使用[21]和[22]的符号和术语。其中，c表示包容，<表示适当包容，当(0)

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Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry

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