t-Linked overrings and prüfer v-multiplication domains

Dobbs, David E.; Houston, Evan; Lucas, Thomas G.; Zafrullah, Muhammad

doi:10.1080/00927878908823879

Cited by 101 publications

(64 citation statements)

References 19 publications

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“…In particular, the classical notion " T is t-linked to D" [6] coincides with the notion " T is t-linked to (D, tD)"…”

Section: Semistar Linkednessmentioning

confidence: 99%

“…In [6], the authors showed that the equality T ℓ t D ,T = T characterizes t-linkedness of T to D. The next goal is to investigate the analogous question in semistar setting.…”

Section: In This Situation Every Principal Ideal Of D Is a Quasi-⋆-imentioning

confidence: 99%

“…In Section 2 we recall the main definitions and we collect some background results on semistar operations. In Section 3, we define and study the notion of semistar linked overring, which generalizes the notion of t-linked overring defined in [6]. Several characterizations of this concept have been obtained.…”

Section: ]: a Domain D Is A Prüfer Domain If And Only If Each Overrinmentioning

confidence: 99%

“…The previous theorems have been extended to the case of Prüfer v-multiplication domains (for short, PvMDs) in [6] and [26], respectively, by means of the v (or the t)-operation.…”

Section: ]: a Domain D Is A Prüfer Domain If And Only If Each Overrinmentioning

confidence: 99%

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Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

Baghdadi

Fontana

2004

Communications in Algebra

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In 1994, Matsuda and Okabe introduced the notion of semistar operation, extending the "classical" concept of star operation. In this paper, we introduce and study the notions of semistar linkedness and semistar flatness which are natural generalizations, to the semistar setting, of their corresponding "classical" concepts. As an application, among other results, we obtain a semistar version of Davis' and Richman's overring-theoretical theorems of characterization of Prüfer domains for Prüfer semistar multiplication domains.

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“…In particular, the classical notion " T is t-linked to D" [6] coincides with the notion " T is t-linked to (D, tD)"…”

Section: Semistar Linkednessmentioning

confidence: 99%

“…In [6], the authors showed that the equality T ℓ t D ,T = T characterizes t-linkedness of T to D. The next goal is to investigate the analogous question in semistar setting.…”

Section: In This Situation Every Principal Ideal Of D Is a Quasi-⋆-imentioning

confidence: 99%

Section: ]: a Domain D Is A Prüfer Domain If And Only If Each Overrinmentioning

confidence: 99%

“…The previous theorems have been extended to the case of Prüfer v-multiplication domains (for short, PvMDs) in [6] and [26], respectively, by means of the v (or the t)-operation.…”

Section: ]: a Domain D Is A Prüfer Domain If And Only If Each Overrinmentioning

confidence: 99%

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Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

Baghdadi

Fontana

2004

Communications in Algebra

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“…It is well-known that D is Archimedean. By [DHLZ,Corollary 2.7], the extension D ⊆ D is R 2 -stable in the sense that whenever 0 = a, b ∈ D and aD ∩ bD = abD, we have aD ∩ bD = abD . By [DZ, Proposition 1], every prime element of D remains prime in D .…”

mentioning

confidence: 99%

Integral Domains Having Nonzero Elements with Infinitely Many Prime Divisors

Coykendall

Dumitrescu

2007

Communications in Algebra

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Abstract. In a factorial domain every nonzero element has only finitely many prime divisors. We study integral domains having nonzero elements with infinitely many prime divisors.Let D be an integral domain. It is well known that if D is a UFD then every nonzero element has only finitely many prime divisors (see e.g. [G]). This is also true if D is a Noetherian domain, or more generally, if D satisfies the ascending chain condition for the principal ideals (ACCP). Indeed, if some nonzero element d ∈ D has infinitely many (mutually non-associate) primes p n , then the principal ideals d/p 1 · · · p n D form a strictly ascending chain. Moreover, the element d cannot be written as a product of irreducible elements, say d = a 1 a 2 · · · a m , because then each p n is an associate of some a j , a contradiction; so D is not even an atomic domain. We call a domain having nonzero elements with infinitely many prime divisors an IPD domain. Examples of IPD domains are not hard to find. For instance, the ring E of entire functions, that is, complex functions which are analytic in the whole plane, is an IPD domain. Indeed, by [G, page 147,, if a, b are distinct complex numbers, then z − a, z − b are non-associated prime elements of E and if (a n ) is an infinite sequence of distinct complex numbers with |a n | → ∞, there exists 0 = f ∈ E divisible by each z − a n . The subring A of Q[X] consisting of all polynomials with constant term in Z is an IPD domain as well, because every prime p ∈ Z is prime in A and p divides X (see also Proposition 2). The IPD domains appear in [MO] under the name of non-GD(1) domains. In [Co], an example of a non-IPD domain whose integral closure is an IPD domain is given.In this note, we study some transfer properties for the IPD domains and indicate some constructions producing IPD domains. First, we describe the IPD domains of type A + XB[X] (Proposition 2). Here, when A ⊆ B is an extension of domains, A + XB[X] is the subring of B[X] consisting of all polynomials f whose constant term is in A. Then, we investigate the possible connections between D being an IPD domain and its integral closure (or its complete integral closure) being an IPD domain (see Propositions 7, 9 and 11).Throughout this paper, all rings are commutative and unitary. For basic results and terminology our reference is [G].Let A ⊆ B be a domain extension and A + XB[X] be the subring of B[X] consisting of all polynomials f whose constant term is in A. Our first aim is to describe the IPD domains of type A + XB[X].2000 Mathematics Subject Classification : 13A15, 13B22, 13F05, 13G05.

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

Fontana¹,

Loper²

Multiplicative Ideal Theory in Commutative Algebra

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t-Linked overrings and prüfer v-multiplication domains

Cited by 101 publications

References 19 publications

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

Integral Domains Having Nonzero Elements with Infinitely Many Prime Divisors

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

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