A Note on Prüfer Semistar Multiplication Domains

Picozza, Giampaolo

doi:10.4134/jkms.2009.46.6.1179

Cited by 3 publications

(1 citation statement)

References 18 publications

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“…An important generalization of the Prüfer domain notion is that of a Prüfer v-multiplication domain (PVMD). This notion comes from multiplicative ideal theory, and various ideal-theoretic properties of it have been considered by many authors, see for example [1,2,4,7,9,10,13,14,15]. From the homological algebra point of view, Prüfer domains are exactly the integral domains of weak global dimension at most one.…”

Section: Introductionmentioning

confidence: 99%

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

Wang¹,

Qiao²

2015

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we introduce and study the w-weak global dimension w-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a Prüfer v-multiplication domain if and only if w-w.gl.dim(R)1. We also show that there is a large class of domains in which Hilbert's syzygy Theorem for the w-weak global dimension does not hold. Namely, we prove that if R is an integral domain (but not a field) for which the polynomial ring R[x] is w-coherent, then w-w.gl.dim(R[x]) = w-w.gl.dim(R).

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Section: Introductionmentioning

confidence: 99%

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

Wang¹,

Qiao²

2015

Bulletin of the Korean Mathematical Society

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On v-domains: a survey

Fontana

Zafrullah

2010

Commutative Algebra

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The v-domains generalize Prüfer and Krull domains and have appeared in the literature with different names. This paper is the result of an effort to put together information on this useful class of integral domains. In this survey, we present old, recent and new characterizations of v-domains along with some historical remarks. We also discuss the relationship of v-domains with their various specializations and generalizations, giving suitable examples.

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On some classes of integral domains defined by Krullʼs a.b. operations

Fontana

Picozza

2011

Journal of Algebra

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Let D be an integral domain with quotient field K. The boperation that associates to each nonzero D-submodule E of K, E b := {EV | V valuation overring of D}, is a semistar operation that plays an important role in many questions of ring theory (e.g., if I is a nonzero ideal in D, I b coincides with its integral closure). In a first part of the paper, we study the integral domains that are b-Noetherian (i.e., such that, for each nonzero ideal I of D, I b = J b for some a finitely generated ideal J of D). For instance, we prove that a b-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily b-Noetherian. We also characterize several distinguished classes of b-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type ⋆a canonically associated to a given semistar operation ⋆ (for instance, the b-operation is the e.a.b. semistar operation of finite type canonically associated to the identity operation). These operations, introduced and studied by Krull, Jaffard, Gilmer and Halter-Koch, play a very important role in the recent generalizations of the Kronecker function ring. In particular, in the present paper, we classify several classes of integral domains having some of the fundamental operations d, t, w and v equal to some of the canonically associated e.a.b. operations b, ta, wa and va.

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A Note on Prüfer Semistar Multiplication Domains

Cited by 3 publications

References 18 publications

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

On v-domains: a survey

On some classes of integral domains defined by Krullʼs a.b. operations

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