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Divisible modules over discrete finite dimension valuation domains. (English) Zbl 0604.13001

The author studies the discrete valuation domains (DVN) of finite Krull dimension, obtaining the structure of divisible modules and of modules whose torsion submodule is divisible.
Main results: Theorem 2.1. Let R be a DVD with \(\dim (R)=n\in {\mathbb{N}}\) and D a divisible R-torsion module. Then inj\(d_ R(D)\leq 1\). - Theorem 2.4. Let R be an almost maximal DVD, \(\dim (R)<\infty\), D a divisible torsion R-module of countable rank. Then D is a direct sum of uniserial modules. - Theorem 3.1. Let R be a maximal DVD, \(\dim (R)<\infty\) and M an R-module such that t(M) is divisible and \(M/d(M)\) is of countable rank. Then \(M\simeq \oplus_{i\in J}R_{{\mathfrak q}_ i}\oplus t(M),\) where \({\mathfrak q}_ i\in Spec(R)\).
Reviewer: M.Ştefănescu

MSC:

13A05 Divisibility and factorizations in commutative rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13B30 Rings of fractions and localization for commutative rings
13F30 Valuation rings
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