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Extensions de valuation et polygone de Newton

Michel Vaquié (2008)

Annales de l’institut Fourier

Soient ( K , ν ) un corps valué et L est une extension monogène finie de K définie par L = K [ x ] / ( P ) , alors toute valuation de L qui prolonge ν définit une pseudo-valuation ζ de K [ x ] de noyau l’idéal ( P ) . Nous savons associer à ζ une famille de valuations de K [ x ] , appelée famille admissible, construite de façon explicite à partir de valuations augmentées et de valuations augmentées limites.Nous donnons une condition nécessaire et suffisante pour qu’une valuation μ de K [ x ] appartienne à la famille admissible associée à une pseudo-valuation...

Formal prime ideals of infinite value and their algebraic resolution

Steven Dale Cutkosky, Samar ElHitti (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

Suppose that R is a local domain essentially of finite type over a field of characteristic 0 , and ν a valuation of the quotient field of R which dominates R . The rank of such a valuation often increases upon extending the valuation to a valuation dominating R ^ , the completion of R . When the rank of ν is 1 , Cutkosky and Ghezzi handle this phenomenon by resolving the prime ideal of infinite value, but give an example showing that when the rank is greater than 1 , there is no natural ideal in R ^ that...

Local monomialization of transcendental extensions

Steven Dale CUTKOSKY (2005)

Annales de l’institut Fourier

Suppose that R ⊂ S are regular local rings which are essentially of finite type over a field k of characteristic zero. If V is a valuation ring of the quotient field K of S which dominates S , then we show that there are sequences of monoidal transforms (blow ups of regular primes) R → R 1 and S → S 1 along V such that R 1 → S 1 is a monomial mapping. It follows that a morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blow ups of nonsingular subvarieties.

Maximal non λ -subrings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

Let R be a commutative ring with unity. The notion of maximal non λ -subrings is introduced and studied. A ring R is called a maximal non λ -subring of a ring T if R ⊂ T is not a λ -extension, and for any ring S such that R ⊂ S ⊆ T , S ⊆ T is a λ -extension. We show that a maximal non λ -subring R of a field has at most two maximal ideals, and exactly two if R is integrally closed in the given field. A determination of when the classical D + M construction is a maximal non λ -domain is given. A necessary condition is given...

Maximal non-Jaffard subrings of a field.

Mabrouk Ben Nasr, Noôman Jarboui (2000)

Publicacions Matemàtiques

A domain R is called a maximal non-Jaffard subring of a field L if R ⊂ L, R is not a Jaffard domain and each domain T such that R ⊂ T ⊆ L is Jaffard. We show that maximal non-Jaffard subrings R of a field L are the integrally closed pseudo-valuation domains satisfying dimv R = dim R + 1. Further characterizations are given. Maximal non-universally catenarian subrings of their quotient fields are also studied. It is proved that this class of domains coincides with the previous class when R is integrally...

On f -rings that are not formally real

James J. Madden (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

Henriksen and Isbell showed in 1962 that some commutative rings admit total orderings that violate equational laws (in the language of lattice-ordered rings) that are satisfied by all totally-ordered fields. In this paper, we review the work of Henriksen and Isbell on this topic, construct and classify some examples that illustrate this phenomenon using the valuation theory of Hion (in the process, answering a question posed in [E]) and, finally, prove that a base for the equational theory of totally-ordered...

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