交换环的惰性极小环扩展从何而来

IF 0.6 Q3 MATHEMATICS

Kyungpook Mathematical Journal Pub Date : 2020-03-31 DOI:10.5666/KMJ.2020.60.1.53

D. Dobbs

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引用次数: 0

摘要

设（A，M）⊂（B，N）是可交换的拟局部环。我们考虑存在环D的性质，使得a⊆D \8838B和扩展D \8834B是惰性的。实例表明，这种D的数目可以是任何非负整数或无穷大。这种D的存在并不意味着M⊆N。此后假设M⊆N。如果域扩展A/M⊆B/N是代数的，则这种D的存在并不意味着B是A上的积分（除非B具有Krull维数0）。如果A/M⊆B/N是极小域扩展，则存在唯一的这样的D，必然由D=A+N给出（但不必是N=MB的情况）。反之亦然，即使M=N和B/M是有限的

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Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from

Let (A,M) ⊂ (B,N) be commutative quasi-local rings. We consider the property that there exists a ring D such that A ⊆ D ⊂ B and the extension D ⊂ B is inert. Examples show that the number of such D may be any non-negative integer or infinite. The existence of such D does not imply M ⊆ N . Suppose henceforth that M ⊆ N . If the field extension A/M ⊆ B/N is algebraic, the existence of such D does not imply that B is integral over A (except when B has Krull dimension 0). If A/M ⊆ B/N is a minimal field extension, there exists a unique such D, necessarily given by D = A+N (but it need not be the case that N = MB). The converse fails, even if M = N and B/M is a finite

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来源期刊

Kyungpook Mathematical Journal MATHEMATICS-

CiteScore

1.30

自引率

0.00%

发文量

期刊介绍： Kyungpook Mathematical Journal is an international journal devoted to significant research concerning all aspects of mathematics. The journal has a preference for papers having a broad interest. One volume of the journal is published every year. Each volume until volume 42 consisted of two issues; however, starting from volume 43(2003), each volume consists of four issues. Authors should strive for expository clarity and good literary style. Manuscripts should be prepared as follows. The first page must consist of a short descriptive title, followed by the name(s) and address(es) of the author(s) along with an electronic address if available.