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Une correspondance entre anneaux partiels et groupes

Published online by Cambridge University Press:  12 March 2014

Patrick Simonetta
Patrick Simonetta*
Affiliation:
Université de Paris VII, Equipe de Logique Mathématique, 2, Place Jussieu, 75005 Paris, France, E-mail: simbaud@logique.jussieu.fr
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Abstract

This work is inspired by the correspondence of Malcev between rings and groups. Let A be a domain with unit, and S a multiplicative group of invertible elements. We define AS as the structure obtained from A by restraining the multiplication to A × S, and σ(AS) as the group of functions from A to A of the form x → xa + b, where (a, b) belongs to S × A. We show that AS and σ(As) are interpretable in each other, and then, that we can transfer some properties between classes (or theories) of “reduced” domains and corresponding groups, such as being elementary, axiomatisability (for classes), decidability, completeness, or, in some cases, existence of a model-completion (for theories).

We study the extensions of the additive group of A by the group S, acting by right multiplication, and show that sometimes σ(AS) is the unique extension of this type. We also give conditions allowing us to eliminate parameters appearing in interpretations.

We emphasize the case where the domain is a division ring K and S is its multiplicative group K×. Here, the interpretations can always be done without parameters. If the centre of K contains more than two elements, then σ(K) is the only extension of the additive group of K by its multiplicative group acting by right multiplication, and the class of all such σ(K)'s is elementary and finitely axiomatisable. We give, in particular, an axiomatisation for this class and for the class of σ(K)'s where K is an algebraically closed field of characteristic 0. From these results it follows that some classical model-companion results about theories of fields can be translated and restated as results about theories of solvable groups of class 2.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 1 , March 1997 , pp. 60 - 78
Copyright
Copyright © Association for Symbolic Logic 1997

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References

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