An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations

Fontana, Marco; Loper, K. Alan

doi:10.1007/978-0-387-36717-0_11

Cited by 31 publications

(16 citation statements)

References 49 publications

(66 reference statements)

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“…Let D be an integrally closed integral domain with quotient field K. where c(f ) is the content of f and ∧ ∆ is the semistar operation defined in Section 2.4. See [15,Chapter 32] or [13] for general properties of Kronecker function rings.…”

Section: An Application: Kronecker Function Ringsmentioning

confidence: 99%

Non-compact subsets of the Zariski space of an integral domain

Spirito¹

2016

Illinois J. Math.

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Let V be a minimal valuation overring of an integral domain D and let Zar(D) be the Zariski space of the valuation overrings of D. Starting from a result in the theory of semistar operations, we prove a criterion under which the set Zar(D) \ {V } is not compact. We then use it to prove that, in many cases, Zar(D) is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.

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Section: An Application: Kronecker Function Ringsmentioning

confidence: 99%

Non-compact subsets of the Zariski space of an integral domain

Spirito¹

2016

Illinois J. Math.

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“…The ring D X N v is a nice generalization of the Nagata ring construction and considered by Kang [17]. We call D X N v the (t-)Nagata ring of D. When no ambiguity may arise, D X N v is simply called the Nagata ring of D. For more on D X and D X N v , PRÜFER-LIKE DOMAINS AND THE NAGATA RING 4247 the reader can be referred to Fontana and Loper's interesting survey article [12] or [11]. In this article, we study when D X and D X N v become globalized pseudovaluation domains.…”

Section: Introductionmentioning

confidence: 98%

PrÜfer-Like Domains and the Nagata Ring of Integral Domains

Chang¹,

Kang²

2011

Communications in Algebra

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A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A → B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common toA and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c f be the ideal of D generated by the coefficients of a polynomial f ∈ D X , N = f ∈ D X c f = D , and N v = f ∈ D X c f −1 = D . In this article, we study when the Nagata ring D X N (more generally, D X N v ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D X N v is a GPVD if and only if D is a t-GPVD and D X N v has Prüfer integral closure, if and only if D X is a t-GPVD, if and only if each overring of D X N v is a GPVD. As a corollary, we have that D X N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We alsogive several examples of integral domains D such that D X N v is a GPVD.

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“…Such operations are also closely related to the theory of Kronecker function rings. For a deeper insight on the recent developements on this topic, see [8], [10], [11], [12], [21], [22].…”

Section: Introductionmentioning

confidence: 99%

Some topological considerations on semistar operations

Finocchiaro¹,

Spirito²

2014

Journal of Algebra

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Abstract. We consider properties and applications of a new topology, called the Zariski topology, on the space SStar(A) of all the semistar operations on an integral domain A. We prove that the set of all overrings of A, endowed with the classical Zariski topology, is homeomorphic to a subspace of SStar(A). The topology on SStar(A) provides a general theory, through which we see several algebraic properties of semistar operation as very particular cases of our construction. Moreover, we show that the subspace SStar f (A) of all the semistar operations of finite type on A is a spectral space.

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An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations

Cited by 31 publications

References 49 publications

Non-compact subsets of the Zariski space of an integral domain

Non-compact subsets of the Zariski space of an integral domain

PrÜfer-Like Domains and the Nagata Ring of Integral Domains

Some topological considerations on semistar operations

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