Topologically defined classes of going-down domains

Papick, Ira J.

doi:10.1090/s0002-9904-1975-13835-3

Cited by 16 publications

(32 citation statements)

References 17 publications

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“…Recall from [34] We will begin with a result from the unpublished doctoral dissertation of M. S. Gilbert [26]. We next give that result and, for the sake of completeness, include Gilbert's proof.…”

Section: Resultsmentioning

confidence: 99%

On Seminormal Integral Domains With Treed Overrings

Dobbs¹

2014

International Electronic Journal of Algebra

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Abstract. The following result uses and generalizes a recent result of Ayache on integrally closed domains. Let R be a commutative integral domain with integral closure R (inside the quotient field K of R) such that each overring of R (inside K) is a treed domain and there exists a finite maximal chain of rings going from R to R . Then R is a seminormal domain if and only if, for each maximal ideal M of R, either R M is a pseudo-valuation domain or, for some positive integer n, there exists a finite maximal chain, of length n, of rings from R M to (R M ) each step of which is (an integral minimal ring extension which is) either decomposed or inert. Examples are given in which the latter option holds where R is one-dimensional and Noetherian.Mathematics Subject Classification (2010): Primary 13B99, 13G05; Secondary 13F05, 13B21, 13E10

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Section: Resultsmentioning

confidence: 99%

On Seminormal Integral Domains With Treed Overrings

Dobbs¹

2014

International Electronic Journal of Algebra

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“…Then it follows by reasoning as in the proof of [33,Proposition 2.12] that T is a going-down domain.…”

Section: Proof It Is Clear That (I) ⇒ (Ii) Conversely Suppose (Ii)mentioning

confidence: 95%

“…(vi) ⇒ (i): As R is a Prüfer domain, it follows easily from [33,Proposition 2.12] that A ⊆ B satisfies the incomparable property, for all overrings A ⊆ B of R. In addition, each integral overring of R is treed, and so we get that each overring of R is treed, by [7,Theorem 5.5]. Let T be an overring of R. Set T o := R ∩ T .…”

Section: Proof It Is Clear That (I) ⇒ (Ii) Conversely Suppose (Ii)mentioning

confidence: 98%

“…AYACHE AND D. E. DOBBS As in [13] and [21], R is said to be a going-down domain if the extension R ⊆ T satisfies the going-down property for each domain T that contains R (as a subring); cf. also [32], [33], [15], [22], [17], [31]. By [21,Theorem 1], the rings T that need to be tested (to check that R ⊆ T satisfies the going-down property) may be restricted to be either valuation overrings of R or simple overrings of R. The most natural examples of going-down domains are arbitrary Prüfer domains and domains of (Krull) dimension at most 1.…”

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Characterizations of the Integral Domains Whose Overrings Are Going-Down Domains

Ayache¹,

Dobbs²

2014

International Electronic Journal of Algebra

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“…An i-domain is an integral domain R such that R → T is an i-morphism for each of its overrings T [28]. Then R is an i-domain amounts to saying that its integral closure R is Prüfer and R → R is an i-morphism [28, Proposition 2.14].…”

Section: Remark 222mentioning

confidence: 99%

Ascending the divided and going-down properties by absolute flatness

Picavet

2012

Arab. J. Math.

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This paper aims to show that the "going-down ring" and the "divided ring" properties ascend along flat morphisms whose co-diagonal morphisms are flat, the so-called absolutely flat morphisms introduced by Olivier. But unibranchedness hypotheses are necessary as any henselization morphism shows. As a by-product, we get that the "unibranched divided ring" property is preserved by the formation of factor domains and by localization with respect to prime ideals. Moreover, we exhibit some ascent results for the "going-down ring" and "divided ring" properties along integral extensions. Mathematics Subject Classification

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Topologically defined classes of going-down domains

Cited by 16 publications

References 17 publications

On Seminormal Integral Domains With Treed Overrings

On Seminormal Integral Domains With Treed Overrings

Characterizations of the Integral Domains Whose Overrings Are Going-Down Domains

Ascending the divided and going-down properties by absolute flatness

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