Equivalence elementaire et decidabilite pour des structures du type groupe agissant sur un groupe abelien

Abstract

AbstractWe prove an Ax-Kochen-Ershov like transfer principle for groups acting on groups. The simplest case is the following: let B be a soluble group acting on an abelian group G so that G is a torsion-free divisible module over the group ring ℤ[B], then the theory of B determines the one of the two-sorted structure 〈G,B,*〉, where * is the action of B on C. More generally, we show a similar principle for structures 〈G,B,*〉, where G is a torsion-free divisible module over the quotient of ℤ[B] by the annulator of G.Two applications come immediately from this result:First, for not necessarily commutative domains, where we consider the action of a subgroup of the invertible elements on the additive group. We obtain then the decidability of a weakened structure of ring, with partial multiplication.The second application is to pure groups. The semi-direct product of G by B is bi-interpretable with our structure 〈G,B,*〉. Thus, we obtain stable decidable groups that are not linear over a field.