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DECIDABILITY FOR THEORIES OF MODULES OVER VALUATION DOMAINS

Published online by Cambridge University Press:  22 April 2015

LORNA GREGORY
LORNA GREGORY*
Affiliation:
THE UNIVERSITY OF MANCHESTER SCHOOL OF MATHEMATICS, OXFORD ROAD MANCHESTER, M13 9PL, UKE-mail:[email protected]
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Abstract

Extending work of Puninski, Puninskaya and Toffalori in [5], we show that if V is an effectively given valuation domain then the theory of all V-modules is decidable if and only if there exists an algorithm which, given a, b ε V, answers whether a ε rad(bV). This was conjectured in [5] for valuation domains with dense value group, where it was proved for valuation domains with dense archimedean value group. The only ingredient missing from [5] to extend the result to valuation domains with dense value group or infinite residue field is an algorithm which decides inclusion for finite unions of Ziegler open sets. We go on to give an example of a valuation domain with infinite Krull dimension, which has decidable theory of modules with respect to one effective presentation and undecidable theory of modules with respect to another. We show that for this to occur infinite Krull dimension is necessary.

Keywords

theory of modulescommutative valuation domaindecidabilityZiegler spectrum
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 2 , June 2015 , pp. 684 - 711
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

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