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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
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##### 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· Zbl 0629.13008 [3] Fuchs, L.; Salce, L., () [4] Prest, M., () [5] M. Prest, Decidability for modules — summary, unpublished notes, 1991 [6] Puninski, G., Cantor – bendixson rank of the ziegler spectrum over a commutative valuation domain, J. symbolic logic, 64, 1512-1518, (1999) ·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
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