[Bi-interpretability and QFA structures: study of some soluble groups and commutative rings]
Une structure S de type fini est dite QFA (pour quasi finiment axiomatisable, voir [A. Nies, Separating classes of groups by first order sentences, Internat. J. Algebra Comput. 13 (2003) 287–302]) s'il existe un énoncé du premier ordre satisfait par S telle que toute structure de type fini qui la satisfait est isomorphe à S. Nous montrons que toute structure bi-interprétable avec l'anneau des entiers est QFA et première. Nous appliquons ce résultat d'une part à certains groupes métabéliens et d'autre part aux anneaux commutatifs.
A finitely generated structure is said to be QFA (for quasi-finitely axiomatizable, see [A. Nies, Separating classes of groups by first order sentences, Internat. J. Algebra Comput. 13 (2003) 287–302]) 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-interprétable 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.
Accepted:
Published online:
Khelif, Anatole 1
@article{CRMATH_2007__345_2_59_0,
author = {Khelif, Anatole},
title = {Bi-interpr\'etabilit\'e et structures {QFA} : \'etude de groupes r\'esolubles et des anneaux commutatifs},
journal = {Comptes Rendus. Math\'ematique},
pages = {59--61},
year = {2007},
publisher = {Elsevier},
volume = {345},
number = {2},
doi = {10.1016/j.crma.2007.06.003},
language = {fr},
url = {https://www.numdam.org/articles/10.1016/j.crma.2007.06.003/}
}
TY - JOUR AU - Khelif, Anatole TI - Bi-interprétabilité et structures QFA : étude de groupes résolubles et des anneaux commutatifs JO - Comptes Rendus. Mathématique PY - 2007 SP - 59 EP - 61 VL - 345 IS - 2 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.crma.2007.06.003/ DO - 10.1016/j.crma.2007.06.003 LA - fr ID - CRMATH_2007__345_2_59_0 ER -
%0 Journal Article %A Khelif, Anatole %T Bi-interprétabilité et structures QFA : étude de groupes résolubles et des anneaux commutatifs %J Comptes Rendus. Mathématique %D 2007 %P 59-61 %V 345 %N 2 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.crma.2007.06.003/ %R 10.1016/j.crma.2007.06.003 %G fr %F CRMATH_2007__345_2_59_0
Khelif, Anatole. Bi-interprétabilité et structures QFA : étude de groupes résolubles et des anneaux commutatifs. Comptes Rendus. Mathématique, Volume 345 (2007) no. 2, pp. 59-61. doi: 10.1016/j.crma.2007.06.003
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