Coherence of Polynomial Rings

Greenberg, Brian; Vasconcelos, Wolmer V.

doi:10.2307/2040750

Cited by 4 publications

(4 citation statements)

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“…Pullbacks were considered in [19] for providing an appropriate unified setting for several important "composite-type" constructions introduced in various contexts of commutative ring theory in order to construct examples and counter-examples with different pathologies: for instance, Seidenberg's constructions for (polynomial) dimensional sequences [43], Nagata's composition of valuation domains and "K + J (R)" constructions [39, p. 35 and Appendix A1, Example 2], Akiba's AV-domains or Dobbs' divided domains [1,16], Gilmer's "D + M" constructions [26], Traverso's glueings for a constructive approach to the seminormalization [44], Vasconcelos' umbrella rings and Greenberg's F-domains [27,45], Boisen-Sheldon's CPI-extensions [13], HedstromHouston's pseudo-valuation domains [29], "D + XD S [X]" rings and more generally, the "A + XB[X]" rings considered by many authors (see the recent excellent survey papers by T. Lucas [35] and M. Zafrullah [46], which contain ample and updated bibliographies on this subject).…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Star operations and pullbacks

Fontana

Park

2004

Journal of Algebra

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In this paper we study the star operations on a pullback of integral domains. In particular, we\ud characterize the star operations of a domain arising froma pullback of “a general type” by introducing\ud new techniques for “projecting” and “lifting” star operations under surjective homomorphisms of\ud integral domains. We study the transfer in a pullback (or with respect to a surjective homomorphism)\ud of some relevant classes or distinguished properties of star operations such as v−, t−, w−, b−,\ud d−, finite type, e.a.b., stable, and spectral operations. We apply part of the theory developed here to\ud give a complete positive answer to a problem posed by D.F. Anderson in 1992 concerning the star\ud operations on the “D +M” constructions

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Star operations and pullbacks

Fontana

Park

2004

Journal of Algebra

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“…The first authors to consider pullback rings with zero divisors were probably Greenberg and Vasconcelos (1974 [31], 1976 [32]). They aimed to classify rings of weak global dimension 2 and considered a very specialized conductor square where i 1 is a flat epimorphism, Q is a flat ideal of R, and QS = S. Fontana (1980 [19]) and Cahen (1988 [13]) also considered a number of properties of pullback rings with zero divisors in a slightly less restrictive setting than the above conductor square.…”

Section: Pullback Ringsmentioning

confidence: 99%

Prüfer Conditions in Commutative Rings

Glaz

Schwarz

2011

Arab J Sci Eng

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This article explores several extensions of the Prüfer domain notion to rings with zero divisors. These extensions include semihereditary rings, rings with weak global dimension less than or equal to 1, arithmetical rings, Gaussian rings, locally Prüfer rings, strongly Prüfer rings, and Prüfer rings. The renewed interest in these properties, due to their connection to Kaplansky's Conjecture, has resulted in a large body of results shedding new light on the area. We survey the work done in this direction in the last 15 years, including results, examples and counterexamples, and a multitude of open problems.

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“…If gl.dim D 2 and w.gl.dim T 2, then R is coherent and w.gl.dim R 2.Proof. By[28, Proposition 4.1], R is coherent. Apply Theorem 4.1.…”

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confidence: 99%

The Bass-Quillen problem on a class of local rings with weak global dimension two

Wang

2009

Sci. China Ser. A-Math.

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Let (R, m) be a local GCD domain. R is called a U2 ring if there is an element u ∈ m − m 2 such that R/(u) is a valuation domain and Ru is a Bézout domain. In this case u is called a normal element of R. In this paper we prove that if R is a U2 ring, then R and R[x] are coherent; moreover, if R has a normal element u and dim(R/(u)) = 1, then every finitely generated projective module over R[X] is free. Keywords: weak global dimension, local domain, projective, free MSC(2000): 13C10, 13D05, 13G05 IntroductionLet R be a commutative ring and R[X 1 , . . . , X n ] be the polynomial ring over R., then M is finitely generated projective over R. It is clear that a free R[X 1 , . . . , X n ]-module is extended from R. By Kaplansky theorem [1] , if a finitely generated projective R[X 1 , . . . , X n ]-module H is extended from R and R is local, then H is free. The problem whether finitely generated projective R[X 1 , . . . , X n ]-modules are extended from R originates from Serre conjecture [2] . In 1955, Serre guessed that, for a field K, each finitely generated projective K[X 1 , . . . , X n ]-module is free. Serre Conjecture was solved independently by Quillen and Suslin in 1976. The problem has an extension to Noetherian regular rings and it is called Bass-Quillen Conjecture (for short, BQ), that is, BQ: Let R be a regular Noetherian local ring. Then every finitely generated projective R[X 1 , . . . , X n ]-module is extended from R.By the Quillen's patching theorem [3] , a finitely presented R[X 1 , . . . , X n ]-module H is extended from R if and only if H m is extended from R m for any maximal ideal m of R. Thus Bass-Quillen Conjecture is equivalent to the following form:BQ d : Let R be a d-dimensional regular Noetherian local ring. Then every finitely generated projective R[X 1 , . . . , X n ]-module is free, where d = gl. dim(R) = dim(R).In the case d 2 Bass-Quillen Conjecture is true (see [4]). For the case d = 3, Rao proved in [5] that Bass-Quillen Conjecture is true if the characteristic is not equal to 2 or 3. Many

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Coherence of Polynomial Rings

Abstract: Abstract. The main result is that A [X], the polynomial ring in any number of indeterminates over a coherent ring A of global dimension two, is coherent.

Cited by 4 publications

References 3 publications

Star operations and pullbacks

Star operations and pullbacks

Prüfer Conditions in Commutative Rings

The Bass-Quillen problem on a class of local rings with weak global dimension two

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