Quasi-Prüfer Extensions of Rings

Picavet, Gabriel; Picavet-L’Hermitte, Martine

doi:10.1007/978-3-319-65874-2_16

Cited by 21 publications

(33 citation statements)

References 46 publications

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“…In [21], a minimal flat epimorphism is called a Prüfer minimal extension. An FCP Prüfer extension has FIP and is a tower of finitely many Prüfer minimal extensions [21,Proposition 1.3].…”

Section: Results On Minimal Extensionsmentioning

confidence: 99%

“…According to [21,Proposition 4.16], R P ⊆ W P is either integral ( * ) or Prüfer ( * * ) for each P ∈ Spec(R). In case ( * ), we get R P = ( R) P ; so that, V P = W P .…”

Section: Co-integral Closuresmentioning

confidence: 99%

“…The Prüfer hull of an extension R ⊆ S is the greatest Prüfer subextension R of [R, S] [19]. In [21], we defined an extension R ⊆ S to be quasi-Prüfer if it can be factored R ⊆ R ′ ⊆ S, where R ⊆ R ′ is integral and R ′ ⊆ S is Prüfer. An FCP extension is quasi-Prüfer [21,Corollary 3.4].…”

Section: Introduction and Notationmentioning

confidence: 99%

“…An extension R ⊆ S is called almost-Prüfer if R ⊆ S is integral, or equivalently, when R ⊆ S is FIP, if S = RR [21,Theorem 4.6]. An almost-Prüfer extension is quasi-Prüfer.…”

Section: Introduction and Notationmentioning

confidence: 99%

See 3 more Smart Citations

Closures and co-closures attached to FCP ring extensions

Picavet¹,

Picavet-L’Hermitte²

2021

Preprint

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The paper deals with ring extensions R ⊆ S and the poset [R, S] of their subextensions, with a special look at FCP extensions (extensions such that [R, S] is Artinian and Noetherian). When the extension has FCP, we show that there exists a co-integral closure, that is a least element R in [R, S] such that R ⊆ S is integral. Replacing the integral property by the integrally closed property, we are able to prove a similar result for an FCP extension. The radicial closure of R in S is well known. We are able to exhibit a suitable separable closure of R in S in case the extension has FCP, and then results are similar to those of field theory. The FCP property being always guaranteed, we discuss when an extension has a co-subintegral or a co-infra-integral closure. Our theory is made easier by using anodal extensions. These (co)-closures exist for example when the extension is catenarian, an interesting special case for the study of distributive extensions to appear in a forthcoming paper.

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Section: Results On Minimal Extensionsmentioning

confidence: 99%

“…According to [21,Proposition 4.16], R P ⊆ W P is either integral ( * ) or Prüfer ( * * ) for each P ∈ Spec(R). In case ( * ), we get R P = ( R) P ; so that, V P = W P .…”

Section: Co-integral Closuresmentioning

confidence: 99%

Section: Introduction and Notationmentioning

confidence: 99%

“…An extension R ⊆ S is called almost-Prüfer if R ⊆ S is integral, or equivalently, when R ⊆ S is FIP, if S = RR [21,Theorem 4.6]. An almost-Prüfer extension is quasi-Prüfer.…”

Section: Introduction and Notationmentioning

confidence: 99%

See 2 more Smart Citations

Closures and co-closures attached to FCP ring extensions

Picavet¹,

Picavet-L’Hermitte²

2021

Preprint

Self Cite

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“…The first part of (2) is [22,Corollary 3.4]. For the second part, consider the tower R ⊆ R ⊆ S whose length is ≤ 2.…”

Section: First Properties Of Extensions Of Lengthmentioning

confidence: 99%

Ring extensions of length two

Picavet

Picavet-L’Hermitte

2019

J. Algebra Appl.

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We characterize extensions of commutative rings R ⊂ S such that R ⊂ T is minimal for each R-subalgebra T of S with T = R, S. This property is equivalent to R ⊂ S has length 2. Such extensions are either pointwise minimal or simple. We are able to compute the number of subextensions of R ⊂ S. Besides commutative algebra considerations, our main result is a consequence of the recently introduced by van Hoeij et al. concept of principal subfields of a finite separable field extension. As a corollary of this paper, we get that simple extensions of length 2 have FIP.2010 Mathematics Subject Classification. Primary:13B02,13B21, 13B22, 12F10; Secondary: 13B30.Key words and phrases. FIP, FCP extension, minimal extension, length of an extension, integral extension, support of a module, t-closure, algebraic field extension, separable field extension, principal subfield, pointwise minimal extension.

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Around Prüfer Extensions of Rings

Picavet

Picavet-L’Hermitte

2023

Algebraic, Number Theoretic, and Topological Aspects of Ring Theory

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Quasi-Prüfer Extensions of Rings

Cited by 21 publications

References 46 publications

Closures and co-closures attached to FCP ring extensions

Closures and co-closures attached to FCP ring extensions

Ring extensions of length two

Around Prüfer Extensions of Rings

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