Intermediary rings in normal pairs

Ayache, Ahmed; Jarboui, Noômen

doi:10.1016/j.jpaa.2008.03.002

Cited by 18 publications

(10 citation statements)

References 7 publications

(9 reference statements)

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“…That is Several papers have provided approximations for the cardinality |[R, S]| in the context of normal pairs with finite supports [3][4][5]12]. Recently, [6] has presented an effective algorithm that enables us to compute the exact value of |[R, S]|.…”

Section: Let a = Supp(s/r) ∪ {O} Ordered By Inclusion Where O Is Thementioning

confidence: 99%

The set of indeterminate rings of a normal pair as a partially ordered set

Ayache

2010

Ricerche mat.

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Let R ⊂ S be an extension of integral domains and let [R, S] be the set of intermediate rings between R and S ordered by inclusion. If (R, S) is normal pair and [R, S] is finite, we do prove that there exists a semi-local Prüfer ring T with quotient field K such that [R, S] ∼ = [T, K ] (as a partially ordered set). Consequently, any problem relative to the finiteness conditions in [R, S] can be investigated in the particular case where R is a semi-local Prüfer ring with quotient field S.

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Section: Let a = Supp(s/r) ∪ {O} Ordered By Inclusion Where O Is Thementioning

confidence: 99%

The set of indeterminate rings of a normal pair as a partially ordered set

Ayache

2010

Ricerche mat.

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“…In this section we collect and recall some results from [4]. Let (R, S) be a normal pair and Supp(S/R) the ordered set…”

Section: Some Basic Properties Of Normal Pairsmentioning

confidence: 99%

“…This paper is a sequel to [4], we adopt the conventions that each ring considered is commutative, with unit and an inclusion (extension) of rings signifies that the smaller ring is a subring of the larger and possesses the same multiplicative identity. Throughout this paper, Spec(R) denotes the set of prime ideals of an integral domain R, and Max(R) denotes the set of its maximal ideals.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for computing the number of intermediary rings in normal pairs

Ayache

Jarboui

2008

Journal of Pure and Applied Algebra

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The main purpose of this paper is to establish a result giving the number of intermediary rings between R and S when (R, S) is a normal pair of rings and to provide an algorithm to compute this number. c 2007 Published by Elsevier B.V. MSC: Primary: 13B02; secondary: 13C15; 13A17; 13A18; 13B25; 13E05 IntroductionThis paper is a sequel to [4], we adopt the conventions that each ring considered is commutative, with unit and an inclusion (extension) of rings signifies that the smaller ring is a subring of the larger and possesses the same multiplicative identity. Throughout this paper, Spec(R) denotes the set of prime ideals of an integral domain R, and Max(R) denotes the set of its maximal ideals. The height of a prime ideal P of R denoted by ht P, is defined to be the supremum of the lengths n of chains of prime ideals of R, P 0 ⊂ P 1 ⊂ · · · ⊂ P n = P. The Krull dimension of R, dim(R), is defined to be the supremum of such heights for P ∈ Max(R). If P ⊆ Q are two prime ideals of R, then [P, Q] will denote the set of prime ideals Q of R such that P ⊆ Q ⊆ Q. For any set X , |X | denotes the cardinality of this set. Let R ⊆ S be an extension of integral domains. The set of subrings of S that contain R is called the set of intermediary rings in the ring extension R ⊆ S. We let [R, S] denote this set. If K is the quotient field of R, then an intermediate ring in the extension R ⊆ K is called an overring of R. If each overring of R is integrally closed in K , then R is called a Prüfer domain (cf. [9]).Recall that a pair of rings (R, S) is called a normal pair provided that R ⊆ S and each T ∈ [R, S] is integrally closed in S. These pairs were first defined and studied by Davis [6]. For example, if R is a Prüfer domain with quotient field K , then (R, K ) is a normal pair. Davis demonstrated that if R is a local ring, then (R, S) is a normal pair if and

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“…He proved that if R is quasilocal, then (R, S ) is a normal pair if and only if there exists a divided prime ideal P of R (that is, PR P = P) such that S = R P and R/P is a valuation ring [7,Theorem 1]. Recently, normal pairs have received considerable attention (see [3,5,9,16] In recent years, there has been increasing interest in ring extensions R ⊂ S satisfying FIP (finite intermediate rings property) or FCP (finite chain property). Following [1], the extension R ⊂ S is said to satisfy FIP if there are only a finite number of rings contained between R and S .…”

Section: Introductionmentioning

confidence: 99%

“…Then[3, Corollary 2.6] ensures that S 1 ⊆ S and R * ⊂ S 1 is a minimal extension with crucial maximal ideal M. As, in addition, R * is the integral closure of J in S 1 and M C, then, according to Proposition 2.5, [J, S 1…”

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When Is the Integral Closure Comparable to All Intermediate Rings

Nasr¹,

Zeidi²

2016

Bull. Aust. Math. Soc.

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Let $R\subset S$ be an extension of integral domains, with $R^{\ast }$ the integral closure of $R$ in $S$. We study the set of intermediate rings between $R$ and $S$. We establish several necessary and sufficient conditions for which every ring contained between $R$ and $S$ compares with $R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150].

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Intermediary rings in normal pairs

Cited by 18 publications

References 7 publications

The set of indeterminate rings of a normal pair as a partially ordered set

The set of indeterminate rings of a normal pair as a partially ordered set

An algorithm for computing the number of intermediary rings in normal pairs

When Is the Integral Closure Comparable to All Intermediate Rings

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