Integer-Valued Polynomials and Prüfer v-Multiplication Domains

Cahen, Paul-Jean; Loper, Alan; Tartarone, Francesca

doi:10.1006/jabr.1999.8155

Cited by 30 publications

(8 citation statements)

References 9 publications

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“…The proof is the same as that given in [9, Proposition 2.5] with the toperation replaced by ⋆ an arbitrary star operation of finite type. The "prime ideal part" of the statement follows from [9 …”

Section: Background and Preliminary Resultsmentioning

confidence: 99%

Nagata Rings, Kronecker Function Rings, and Related Semistar Operations

Fontana

Loper

2003

Communications in Algebra

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Abstract. In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer's book [20]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer and P. Lorenzen from 1930's.In [17] and [18] the current authors investigated properties of the Kronecker function rings which arise from arbitrary semistar operations on an integral domain D. In this paper we extend that study and also generalize Kang's notion of a star Nagata ring [30] and [31] to the semistar setting. Our principal focuses are the similarities between the ideal structure of the Nagata and Kronecker semistar rings and between the natural semistar operations that these two types of function rings give rise to on D.

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Section: Background and Preliminary Resultsmentioning

confidence: 99%

Nagata Rings, Kronecker Function Rings, and Related Semistar Operations

Fontana

Loper

2003

Communications in Algebra

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“…Moreover, since Z ultra is compact (by (4)) and Z # is Hausdorff (by (3)), id Z is a closed map (cf., for instance, [8, Chapter IX, Theorem 2.1]), and hence is a homeomorphism (cf., for instance, [8, Chapter III, Theorem 12.2]). Finally, the equality Z # = Z cons follows immediately from (4) and from the definition of the constructible topology.…”

Section: By Lemma 21(3) We Want To Show Thatmentioning

confidence: 94%

“…This notion of ultrafilter limit points of collections of prime ideals has been used to great effect in several recent papers [4], [21], and [22]. If U is a trivial ultrafilter on C then, by definition, there is a prime P ∈ C such that U = {Z ∈ B(C) | P ∈ Z} (=: β P C ) and it is straightforward in this case that P U = P ∈ C [13, page 2918].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

Finocchiaro

Fontana

Loper

2013

Communications in Algebra

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Abstract. Let K be a field and let A be a subring of K. We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on the space Zar(K|A) of all valuation domains having K as quotient field and containing A. We show that the ultrafilter topology coincides with the constructible topology on the abstract Riemann-Zariski surface Zar(K|A). We extend results regarding distinguished spectral topologies on spaces of valuation domains.

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“…If U ∈ βΣ, then the set Σ U := {x ∈ X : Σ(x) ∈ U } is usually called the ultrafilter limit point of Σ, with respect to U . It is easy to see that Σ U is an ideal of A (see [1]). (i) The set Spec(A) of all prime ideals of A (see [1] or [4]).…”

Section: Ultrafilter Limit Pointsmentioning

confidence: 98%

“…(ii) The set I(A) of all ideals of A (see [1]). The following result will be used often in the sequel.…”

Section: Ultrafilter Limit Pointsmentioning

confidence: 99%

The strong ultrafilter topology on spaces of ideals

Finocchiaro¹,

Loper²

2016

Journal of Algebra

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The patch/constructible refinement of the Zariski topology on the prime spectrum of a commutative ring is well known and well studied. Recently, Fontana and Loper gave an equivalent definition of this topology using ultrafilters. In this note we distinguish between two different types of ultrafilter convergence and use them to define two new topologies on the prime spectrum of a ring. We study various properties of these topologies. As applications we use the ultrafilters to classify all the compact subsets of a spectral space in the Zariski topology and we classify Grothendieck's retrocompact spaces again using ultrafilters.

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Integer-Valued Polynomials and Prüfer v-Multiplication Domains

Abstract: Ž .Ä Let D be a domain with quotient field K. We consider the ring Int D [ f g w x Ž .x K X ; f D : D of integer-valued polynomial rings over D. We completely Ž . characterize the domains D for which Int D is a Prufer¨-multiplication domain.

Cited by 30 publications

References 9 publications

Nagata Rings, Kronecker Function Rings, and Related Semistar Operations

Nagata Rings, Kronecker Function Rings, and Related Semistar Operations

Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

The strong ultrafilter topology on spaces of ideals

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