Prüfer v-multiplication domains and the ring R[X]Nv

Kang, Byung Gyun

doi:10.1016/0021-8693(89)90040-9

Cited by 218 publications

(119 citation statements)

References 11 publications

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“…G. Kang proved that R is a PvMD if and only if R X is a Prüfer (indeed a Bezout) domain [30,Theorem 3.7]. In addition, there is a lattice isomorphism between the lattice of t-ideals of R and the lattice of ideals of R X [30,Theorem 3.4].…”

Section: T##-property and T-radical Trace Propertymentioning

confidence: 99%

“…In fact each Noetherian domain has the t##-property An integral domain R is an almost Krull domain if R M is a rank-one discrete valuation domain for each t-maximal ideal M of R. Almost Krull domains were studied by Kang under the name of t-almost Dedekind domains in [30,Section IV]. A Krull domain is an almost Krull domain with t-finite character.…”

Section: Theorem 112 Let R Be a Pvmd And Consider The Following Conmentioning

confidence: 99%

“…Thus in general it is often more convenient to consider the w-analogue of a given property (see for instance [43,44,9]). However in a PvMD the w-operation and the t-operation coincide [30] and one can use indifferently these two star operations.…”

Section: Introductionmentioning

confidence: 99%

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Ring-Theoretic Properties of PvMDs

Baghdadi

Gabelli

2007

Communications in Algebra

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Section: T##-property and T-radical Trace Propertymentioning

confidence: 99%

Section: Theorem 112 Let R Be a Pvmd And Consider The Following Conmentioning

confidence: 99%

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Ring-Theoretic Properties of PvMDs

Baghdadi

Gabelli

2007

Communications in Algebra

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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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“…(Note: We do assume for both of these results that we know how to construct the Nagata ring R(X) for a quasilocal domain R; in this situation, the condition "c(g) is invertible" becomes "c(g) is principal", [14 Both the Kronecker function ring and the Nagata ring have been generalized and intensively studied (cf. for instance [22], [3], [4], [1], [19], [27], [9], [11] and [17]). However, in spite of their common origin, they have been studied separately.…”

Section: Introductionmentioning

confidence: 99%

A generalization of Kronecker function rings and Nagata rings

Fontana

Loper

2007

Forum Mathematicum

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Abstract. Let D be an integral domain with quotient field K. The Nagata ring D(X) and the Kronecker function ring Kr(D) are both subrings of the field of rational functions K(X) containing as a subring the ring D[X] of polynomials in the variable X. Both of these function rings have been extensively studied and generalized. The principal interest in these two extensions of D lies in the reflection of various algebraic and spectral properties of D and Spec(D) in algebraic and spectral properties of the function rings. Despite the obvious similarities in definitions and properties, these two kinds of domains of rational functions have been classically treated independently, when D is not a Prüfer domain. The purpose of this note is to study two different unified approaches to the Nagata rings and the Kronecker function rings, which yield these rings and their classical generalizations as special cases.

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Prüfer v-multiplication domains and the ring R[X]Nv

Cited by 218 publications

References 11 publications

Ring-Theoretic Properties of PvMDs

Ring-Theoretic Properties of PvMDs

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

A generalization of Kronecker function rings and Nagata rings

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