The class semigroup of local one-dimensional domains

Zanardo, Paolo

doi:10.1016/j.jpaa.2008.03.015

Cited by 12 publications

(8 citation statements)

References 30 publications

(70 reference statements)

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“…Besides, the intersection n≥0 p n of all powers of the maximal ideal p in R coincides with p, while one has n≥0 p n = 0 in any almost perfect local domain [14,Corollary 4.2]. Another example of a one-dimensional local domain that is not almost perfect can be found in [16,Example 1.3]. In view of Corollary 4.12, these are examples of non-CFQ domains.…”

Section: Non-noetherian Ringsmentioning

confidence: 99%

Countably generated flat modules are quite flat

Hrbek,

Positselski,

Slávik

2019

Preprint

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We prove that if R is a commutative Noetherian ring, then every countably generated flat R-module is quite flat, i.e., a direct summand of a transfinite extension of localizations of R in countable multiplicative subsets. We also show that if the spectrum of R is of cardinality less than κ, where κ is an uncountable regular cardinal, then every flat R-module is a transfinite extension of flat modules with less than κ generators. This provides an alternative proof of the fact that over a commutative Noetherian ring with countable spectrum, all flat modules are quite flat. Over a non-Noetherian commutative ring, countably presented flat modules do not need to be quite flat; still over von Neumann regular rings, S-almost perfect rings, and over strongly discrete valuation domains they are.

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Section: Non-noetherian Ringsmentioning

confidence: 99%

Countably generated flat modules are quite flat

Hrbek,

Positselski,

Slávik

2019

Preprint

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“…By construction, Note that the characterization of Boolean regular valuation domains given in [19,Lemma 3.5] is not correct (valuation domains with value group R are the simplest counterexample; this was also observed in [22]). A non-empty proper subset U Γ D is called a filter if Γ D (α) ⊂ U for all α ∈ U.…”

Section: Is Boolean Regular If and Only If γ D Is Algebraically Commentioning

confidence: 99%

Clifford semigroups of ideals in monoids and domains

Halter‐Koch¹

2009

Forum Mathematicum

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We investigate the ideal semigroup and the ideal class semigroup built by the fractional ideals of an ideal system on a monoid or on a domain. We provide criteria for these semigroups to be Clifford semigroups or Boolean semigroups. In particular, we consider the case of valuation monoids (domains) and of Prüfer-like monoids (domains). By the way, we prove that a monoid (domain) is of Krull type if every locally principal ideal is finite. 2000 Mathematics Subject Classification: 13C18, 13F05, 20M12, 20M14, 20M25. 2. By [12, Proposition 4.8.2]. 3. It suffices to prove that J is r-locally principal. Let M ∈ r-max(D). If J ⊂ M , then J M = D M . If J ⊂ M , then Q M ∩ D = Q implies z / ∈ Q M , hence Q M ⊂ zD M and J M = zD M .

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“…The main results regarding the connection between stable rings and the generic formal fiber are in Sections 5 and 6, but since our main focus is on 1-dimensional stable rings, and since stable rings are also of interest in non-Noetherian commutative ring theory (for some recent examples, see [4], [6], [9], [19], [28], [32], [33], [34]), we include in Sections 3 and 4 characterizations of these rings in terms of their normalization and completion; for more in this vein, see also [26].…”

Section: §1 Introductionmentioning

confidence: 99%

Generic formal fibers and analytically ramified stable rings

Olberding¹

2013

Nagoya Mathematical Journal

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Abstract. Let A be a local Noetherian domain of Krull dimension d. Heinzer, Rotthaus and Sally have shown that if the generic formal fiber of A has dimension d − 1, then A is birationally dominated by a one-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of A. We explore further this correspondence between prime ideals in the generic formal fiber and one-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

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The class semigroup of local one-dimensional domains

Cited by 12 publications

References 30 publications

Countably generated flat modules are quite flat

Countably generated flat modules are quite flat

Clifford semigroups of ideals in monoids and domains

Generic formal fibers and analytically ramified stable rings

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