Fuchs, Laszlo; Salce, Luigi S-divisible modules over domains. (English) Zbl 0767.13001 Forum Math. 4, No. 4, 383-394 (1992). 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 \(R_ S/R\). Reviewer: S.Balcerzyk (Toruń) Cited in 3 ReviewsCited in 11 Documents MSC: 13A05 Divisibility and factorizations in commutative rings 13C05 Structure, classification theorems for modules and ideals in commutative rings 13G05 Integral domains Keywords:divisible modules; divisible hull; projective dimension; valuation domain PDFBibTeX XMLCite \textit{L. Fuchs} and \textit{L. Salce}, Forum Math. 4, No. 4, 383--394 (1992; Zbl 0767.13001) Full Text: DOI EuDML