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Gregory, Lorna 
ORCID: https://orcid.org/0000-0002-5508-7217
(2015)
Decidability for theories of modules over valuation domains.
Journal of Symbolic Logic, 80 (2).
pp. 684-711.
ISSN 0022-4812
Abstract
Extending work of Puninski, Puninskaya andToffalori in [5],we showthat ifV 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 ∈ V, answers whether a ∈ 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.
| Item Type: | Article |
|---|---|
| Additional Information: | Publisher Copyright: © 2015, Association for Symbolic Logic. |
| Uncontrolled Keywords: | commutative valuation domain,decidability,theory of modules,ziegler spectrum,philosophy,logic ,/dk/atira/pure/subjectarea/asjc/1200/1211 |
| Faculty \ School: | Faculty of Science > School of Mathematics (former - to 2024) |
| UEA Research Groups: | Faculty of Science > Research Groups > Algebra, Logic & Number Theory |
| Related URLs: | |
| Depositing User: | LivePure Connector |
| Date Deposited: | 17 Oct 2022 12:30 |
| Last Modified: | 07 Nov 2024 12:45 |
| URI: | https://ueaeprints.uea.ac.uk/id/eprint/89130 |
| DOI: | 10.1017/jsl.2014.1 |
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