An algorithmic approach to the existence of ideal objects in commutative algebra

The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert's program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel's no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical

Évolution de la contribution française à l'upgrade de LHCb

Ce document décrit l'évolution de la contribution française à l'upgrade de LHCb. Il s'inscrit dans le prolongement de la Lettre d'Intention [1], du Framework TDR [2], du document soumis au Conseil scientifique de l'IN2P3 le 21 juin 2012 [3], et des Technical Design Reports soumis au LHCC en novembre 2013 [4, 5]. Ces derniers concernent le détecteur de vertex et les détecteurs utilisés dans l'identification des particules. La contribution française s'est cristallisée autour de quatre grands projets : l'électronique front-end des calorimètres et du trajectographe à fibres scintillantes, le système de déclenchement de premier niveau et la carte de lecture à 40MHz commune à l'ensemble des sous-systèmes. Dans ce document nous décrivons les contributions envisagées et les ressources nécessaires pour mener à bien ces projets

The Gr\uf6bner ring conjecture in one variable

We prove that a valuation domain V has Krull dimension 64 1 if and only if for every finitely generated ideal I of V[X] the ideal generated by the leading terms of elements of I is also finitely generated. This proves the Gr\uf6bner ring conjecture in one variable. The proof we give is both simple and constructive. The same result is valid for semihereditary rings

An Algorithmic Approach to the Existence of Ideal Objects in Commutative Algebra

The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn\u2019s lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert\u2019s program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel\u2019s no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical

The Computational Significance of Hausdorff\u2019s Maximal Chain Principle

As a fairly frequent form of the Axiom of Choice about relatively simple structures (posets), Hausdorff\u2019s Maximal Chain Principle appears to be little amenable to computational interpretation. This received view, however, requires revision. When attempting to convert Hausdorff\u2019s principle into a conservation theorem, we have indeed found out that maximal chains are more reminiscent of maximal ideals than it might seem at first glance. The latter live in richer algebraic structures (rings), and thus are readier to be put under computational scrutiny. Exploiting the newly discovered analogy between maximal chains and ideals, we can carry over the concept of Jacobson radical from a ring to an arbitrary set with an irreflexive symmetric relation. This achievement enables us to present a generalisation of Hausdorff\u2019s principle first as a semantic and then as a syntactical conservation theorem. We obtain the latter, which is nothing but the desired computational core of Hausdorff\u2019s principle, by passing from maximal chains to paths of finite binary trees of an adequate inductively generated class. In addition to Hausdorff\u2019s principle, applications include the Maximal Clique Principle for undirected graphs. Throughout the paper we work within constructive set theory