Journal of Symbolic Logic

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

Patrick Simonetta

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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

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745631

Mathematical Reviews number (MathSciNet)
MR1665714

Zentralblatt MATH identifier
0927.03066

JSTOR
links.jstor.org

Citation

Simonetta, Patrick. Equivalence Elementaire et Decidabilite pour des Structures du Type Groupe Agissant sur un Groupe Abelien. J. Symbolic Logic 63 (1998), no. 4, 1255--1285. https://projecteuclid.org/euclid.jsl/1183745631


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