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2016-10-21 - Colloque/Présentation - communication orale - Anglais - 21 page(s)

Point Françoise , "Decidability questions for theories of modules over certain Bézout rings" in BN-pair, a conference in the honor of the 60th birthdays of Alexandre Borovik and Ali Nesin, Istanboul, Turquie, 2016

  • Codes CREF : Algèbre - théorie des anneaux - théorie des corps (DI1147), Logique mathématique (DI1170)
  • Unités de recherche UMONS : Logique mathématique (S838)
  • Instituts UMONS : Institut de Recherche sur les Systèmes Complexes (Complexys)
  • Centres UMONS : Algèbre, Géométrie et Interactions fondamentales (AGIF)

Abstract(s) :

(Anglais) Title: Decidability questions for theories of modules over certain Bézout domains. We introduce a notion of $\ell$-valued modules over a commutative Bézout domain $B$, an instance being the ring $B$ endowed with the $\ell$-valuation map to its lattice-ordered group of divisibility. This valuation map endows any $B$-module with a lattice of subgroups. Restricting ourselves to this abelian reduct, we obtain for those which satisfy a divisibility condition, a relative quantifier elimination result. One ingredient is a Feferman-Vaught type theorem for these $\ell$-valued modules, which allows us to work over the localizations $B_{\M}$, where $\M$ ranges over the maximal spectrum of $B$. We derive decidability results for the theory of modules over certain countable effectively given B\'ezout domains with good factorisation and in particular for countable effectively given {\it good Rumely} domains, for instance the rings $\tilde \Z$, $\tilde \Z\cap \R$ or $\tilde \Z\cap \Q_{p}$, where $\tilde \Z$ denotes the ring of algebraic integers. \par This is joint work with Sonia L'Innocente (University of Camerino).