## Journal of Symbolic Logic

### Equivalence Elementaire et Decidabilite pour des Structures du Type Groupe Agissant sur un Groupe Abelien

Patrick Simonetta

#### Abstract

We 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 $\mathbb{Z}$[B], then the theory of B determines the one of the two-sorted structure $\langle G, B, *\rangle$, where * is the action of B on G. More generally, we show a similar principle for structures $\langle G, B, *\rangle$, where G is a torsion-free divisible module over the quotient of $\mathbb{Z}$[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 $\langle G, B, *\rangle$. Thus, we obtain stable decidable groups that are not linear over a field.

#### Article information

Source
J. Symbolic Logic, Volume 63, Issue 4 (1998), 1255-1285.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745631

Mathematical Reviews number (MathSciNet)
MR1665714

Zentralblatt MATH identifier
0927.03066

JSTOR