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Elimination of quantifiers for ordered valuation rings

Published online by Cambridge University Press:  12 March 2014

M. A. Dickmann
M. A. Dickmann*
Affiliation:
Equipe de Logique Mathématique, U. E. R. De Mathématique et Informatique, Universite Paris VII, Paris, France
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Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result:

Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that:

  • (1) The symbol “∣” is interpreted as the honest divisibility relation:

  • (2) The following divisibility property holds in T:

If T admits q.e. in ℒ, then T = RCVR.

We do not know at present whether the restriction imposed by condition (2) can be weakened.

The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings, for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings:

Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent:

  • (a) A is a convexly ordered valuation ring.

  • (b) Every ideal (or, equivalently, principal ideal) is convex in A.

  • (c) A is a valuation ring convex in its field of fractions quot(A).

  • (d) A is a valuation ring and its maximal ideal MA is convex (in A or, equivalently, in quot (A)).

  • (e) A is a valuation ring and its maximal ideal is bounded by ± 1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 1 , March 1987 , pp. 116 - 128
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[1] Becker, T., Real closed rings and ordered valuation rings, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 29 (1983), pp. 417–425.CrossRefGoogle Scholar
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