caa a22 4500 388038926 CHVBK 20180307125015.0 cr unu---uuuuu 161130e199812 xx s 000 0 eng 10.2307/2586650 doi S0022481200014250 pii (NATIONALLICENCE)cambridge-10.2307/2586650 Simonetta Patrick Université de Paris 7. Laboratoire de Logique, 2. Place Jussieu, 75005 Paris, France. E-mail: simbaud@logique.jussieu.fr Equivalence elementaire et decidabilite pour des structures du type groupe agissant sur un groupe abelien [Elektronische Daten] [Patrick Simonetta] 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 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. Copyright © Association for Symbolic Logic 1998 The Journal of Symbolic Logic Cambridge University Press 63/4(1998-12), 1255-1285 0022-4812 63:4<1255 1998 63 JSL https://doi.org/10.2307/2586650 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586650 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Simonetta Patrick Université de Paris 7. Laboratoire de Logique, 2. Place Jussieu, 75005 Paris, France. E-mail: simbaud@logique.jussieu.fr NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/4(1998-12), 1255-1285 0022-4812 63:4<1255 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge