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Approximate roots of a valuation and the Pierce-Birkhoff conjecture

F. Lucas, J. Madden, D. Schaub, M. Spivakovsky (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper, we construct an object, called a system of approximate roots of a valuation, centered in a regular local ring, which describes the fine structure of the valuation (namely, its valuation ideals and the graded algebra). We apply this construction to valuations associated to a point of the real spectrum of a regular local ring A . We give two versions of the construction: the first, much simpler, in a special case (roughly speaking, that of rank 1 valuations), the second – in the case...

Archimedean frames, revisited

Jorge Martinez (2008)

Commentationes Mathematicae Universitatis Carolinae

This paper extends the notion of an archimedean frame to frames which are not necessarily algebraic. The new notion is called joinfitness and is Choice-free. Assuming the Axiom of Choice and for compact normal algebraic frames, the new and the old coincide. There is a subfunctor from the category of compact normal frames with skeletal maps with joinfit values, which is almost a coreflection. Conditions making it so are briefly discussed. The concept of an infinitesimal element arises naturally,...

Arithmetic of non-principal orders in algebraic number fields

Andreas Philipp (2010)

Actes des rencontres du CIRM

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

Associated graded rings and connected sums

H. Ananthnarayan, Ela Celikbas, Jai Laxmi, Zheng Yang (2020)

Czechoslovak Mathematical Journal

In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ring Q , which force it to be a connected sum over its residue field. In particular, we recover some results regarding short, and stretched, Gorenstein Artin rings. Finally, using these decompositions, we obtain results about the rationality of the Poincaré...

Associated primes of local cohomology modules of generalized Laskerian modules

Dawood Hassanzadeh-Lelekaami, Hajar Roshan-Shekalgourabi (2019)

Czechoslovak Mathematical Journal

Let ℐ be a set of ideals of a commutative Noetherian ring R . We use the notion of ℐ -closure operation which is a semiprime closure operation on submodules of modules to introduce the class of ℐ -Laskerian modules. This enables us to investigate the set of associated prime ideals of certain ℐ -closed submodules of local cohomology modules.

Atomicity and the fixed divisor in certain pullback constructions

Jason Greene Boynton (2012)

Colloquium Mathematicae

Let D be an integral domain with field of fractions K. In this article, we use a certain pullback construction in the spirit of Int(E,D) that furnishes many examples of domains between D[x] and K[x] in which there are elements that do not admit a finite factorization into irreducible elements. We also define the notion of a fixed divisor for this pullback construction to characterize all of its irreducible elements and those nonzero nonunits that do admit a finite factorization into irreducibles....

Avoidance principle and intersection property for a class of rings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

Let R be a commutative ring with identity. If a ring R is contained in an arbitrary union of rings, then R is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in R , then R contains one of them under various conditions.

Calculs d'invariants primitifs de groupes finis

Ines Abdeljaouad (2010)

RAIRO - Theoretical Informatics and Applications

We introduce in this article a new method to calculate all absolute and relatif primitive invariants of finite groups. This method is inspired from K. Girstmair which calculate an absolute primitive invariant of minimal degree. Are presented two algorithms, the first one enable us to calculate all primitive invariants of minimal degree, and the second one calculate all absolute or relative primitive invariants with distincts coefficients. This work take place in Galois Theory and Invariant Theory. ...

Cale Bases in Algebraic Orders

Martine Picavet-L’Hermitte (2003)

Annales mathématiques Blaise Pascal

Let R be a non-maximal order in a finite algebraic number field with integral closure R ¯ . Although R is not a unique factorization domain, we obtain a positive integer N and a family 𝒬 (called a Cale basis) of primary irreducible elements of R such that x N has a unique factorization into elements of 𝒬 for each x ∈ R coprime with the conductor of R . Moreover, this property holds for each nonzero x ∈ R when the natural map Spec ( R ¯ ) → Spec ( R ) is bijective. This last condition is actually equivalent to several properties linked...

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