×

zbMATH — the first resource for mathematics

Decidability of the theory of modules over commutative valuation domains. (English) Zbl 1111.03011
A commutative ring \(V\) with unity is said to be a valuation ring if the lattice of ideals of \(V\) is a chain. A valuation ring without zero divisors is called a valuation domain. It is proved that if \(V\) is an effectively given valuation domain such that its value group is dense and archimedean, then the theory of all (unitary) \(V\)-modules is decidable.

MSC:
03B25 Decidability of theories and sets of sentences
13A18 Valuations and their generalizations for commutative rings
13G05 Integral domains
13C99 Theory of modules and ideals in commutative rings
03C60 Model-theoretic algebra
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Eklof, P.C.; Fischer, E., Elementary theory of abelian groups, Ann. math. logic, 4, 115-171, (1972) · Zbl 0248.02049
[2] Facchini, A., Relative injectivity and pure-injective modules over Prüfer rings, J. algebra, 110, 380-406, (1987) · Zbl 0629.13008
[3] Fuchs, L.; Salce, L., ()
[4] Prest, M., ()
[5] M. Prest, Decidability for modules — summary, unpublished notes, 1991
[6] Puninski, G., Cantor – bendixson rank of the ziegler spectrum over a commutative valuation domain, J. symbolic logic, 64, 1512-1518, (1999) · Zbl 0961.03036
[7] Puninski, G., Serial rings, (2001), Kluwer · Zbl 1032.16001
[8] Szmielew, W., Elementary properties of abelian groups, Fund. math., 41, 203-271, (1955) · Zbl 0064.00803
[9] Ziegler, M., Model theory of modules, Ann. pure appl. logic, 26, 149-213, (1984) · Zbl 0593.16019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.