×

Bi-interpretability and QFA structures: study of some soluble groups and commutative rings. (Bi-interprétabilité et structures QFA : étude de groupes résolubles et des anneaux commutatifs. (French. English summary) Zbl 1117.03041

Summary: A finitely generated structure \(S\) is said to be QFA (for quasi-finitely axiomatizable, see [A. Nies, “Separating classes of groups by first-order sentences”, Int. J. Algebra Comput. 13, 287–302 (2003; Zbl 1059.20002)]) if there exists a first-order sentence satisfied by \(S\) such that every finitely generated structure satisfying it is isomorphic to \(S\). We prove that every structure which is bi-interpretable with the ring of integers is QFA and prime. We apply this result, on the one hand, to some metabelian groups and, on the other, to commutative rings.

MSC:

03C60 Model-theoretic algebra
20A15 Applications of logic to group theory
13L05 Applications of logic to commutative algebra

Citations:

Zbl 1059.20002
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Belegradek, O. V., Model theory of unitriangular groups, (Amer. Math. Soc. Transl., vol. 195 (1999)), 1-116 · Zbl 0931.03053
[2] Hodges, W., Model Theory, Encyclopedia of Mathematics and its Applications (1993), Cambridge University Press
[3] Nies, A., Separating classes of groups by first order sentences, Internat. J. Algebra Comput., 13, 287-302 (2003) · Zbl 1059.20002
[4] A. Nies, Describing groups, Bull. Symb. Logic, à paraître; A. Nies, Describing groups, Bull. Symb. Logic, à paraître
[5] F. Oger, Some new examples of quasi-finitely axiomatizable groups which are prime models, Preprint; F. Oger, Some new examples of quasi-finitely axiomatizable groups which are prime models, Preprint · Zbl 1121.20001
[6] Romanovskii, N. S.; Timoshenko, E. I., On some elementary properties of soluble groups of derived length 2, Sib. Math. J., 44, 2, 350-354 (2003)
[7] T. Scanlon, Infinite finitely generated fields are biinterpretable with N, Preprint; T. Scanlon, Infinite finitely generated fields are biinterpretable with N, Preprint · Zbl 1209.12008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.