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S-divisible modules over domains. (English) Zbl 0767.13001

A multiplicative subset \(S\) of a commutative domain \(R\) determines two classes of \(R\)-modules:
(1) \(S\)-divisible modules \(M\), i.e. such that \(sM = M\) for \(s\in S\),
(2) \(h_ S\)-divisible modules, defined as epimorphic images of direct sums of copies of the localization \(R_ S\) of \(R\).
A generator \(\partial_ S\) for the category of \(S\)-divisible \(R\)-modules is constructed as an \(R\)-module generated by all finite sequences of elements of \(S\) with relations \(s_ k(s_ 1,\dots,s_ k) = (s_ 1,\dots,s_{k-1})\), \(k \geq 1\). Main result: The following conditions are equivalent:
(i) \(\partial_ S\) is an \(h_ S\)-divisible module, (ii) all \(S\)- divisible \(R\)-modules are \(h_ S\)-divisible.
There are also results on a direct decomposition of \(R_ S/R\).

MSC:

13A05 Divisibility and factorizations in commutative rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
13G05 Integral domains
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