Divisible modules over integral domains
The aim of this paper is to describe an equivalence between the full sub- category of Mod-R whose objects are the divisible modules over an integral domain R and a suitable full subcategory of modules over the endomorphism ring E of a fixed divisible module \(\partial.\) This equivalence corresponds to similar equivalences for torsion divisible abelian groups due to \textit{D. K. Harrison} [Ann. Math., II. Ser. 69, 366-391 (1959; Zbl 0100.02901)], and for torsion h-divisible modules over an integral domain due to \textit{E. Matlis} [cf. Mem. Am. Math. Soc. 49 (1964; Zbl 0135.078) and the books: ``Torsionfree modules'', Chicago Lect. Math. (1972; Zbl 0298.13001), and ``1-dimensional Cohen- Macaulay rings'', Lect. Notes Math. 327 (1973; Zbl 0264.13012)]. Let R be a commutative integral domain \((1\in R)\), not a field, and \(\partial_ R\) the divisible right R-module defined by \textit{L. Fuchs} [in: Abelian groups and modules, Proc. Conf., Udine 1984, CISM Courses Lect. 287, 341- 356 (1984; Zbl 0574.13001)]. If \(E=End_ R(\partial_ R)\) and \(\partial\) is viewed as a left \(E\)-module \(_ E\partial\), then \(End(_ E\partial)\simeq R\), and \({}_ E\partial \simeq E/I\) for a suitable projective principal left ideal I of R. Moreover, \(\partial\) has flat and projective dimensions equal to one, both as a right R-module and a left E-module. This implies that the class \({\mathcal F}\) of all right E-modules M such that \(Tor^ E_ 1(M;\partial)=0\) is the torsionfree class for a (non-hereditary) torsion theory (\({\mathcal T};{\mathcal F})\) in Mod-E. A module \(M_ E\) is I-divisible if the canonical homomorphism \(M\otimes_ EI\to M\) is an epimorphism, and that a right E-module \(N_ E\) is I-reduced if it is cogenerated by the right E-module \(\partial^*=\Hom_ R(\partial;C)\), where C is the minimal injective cogenerator in Mod-R. A module \(M_ E\) is I-divisible iff \(\Hom(M;N)=0\) (for all I-reduced \(N_ E)\). A right E-module M is called I-cotorsion if it is I-reduced and \(Ext^ 1_ E(N;M)=0\) (for all I-divisible, I-torsionfree modules \(N_ E).\) Main result: The functors \(\Hom_ R(\partial;-):\quad Mod-R\to Mod-E,\) and \(-\otimes_ E\partial:Mod-E\to Mod-R\) induce an equivalence between the full subcategory of Mod-R whose objects are the divisible R-modules and the full sub-category of Mod-E whose objects are the I-cotorsion E-modules.
divisible modules over an integral domain
scientific article; zbMATH DE number 4067051
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