\(h\)-divisible and cotorsion modules over one-dimensional Gorenstein rings
The aim of this paper is to study the class \(\mathcal C\) of all commutative Noetherian rings whose self injective dimension (id) is at most one. Results from Dedekind rings are extended in this frame. Let \(R\) be a commutative Noetherian ring and \(M\) an \(R\)-module. \(M\) is \(h\)-divisible if it is a homomorphic image of an injective \(R\)-module. A characterization of \(\mathcal C\) in terms of \(h\)-divisibility is given. \(M\) is said to be cotorsion if \(\text{Ext}_ R^ 1(F,M)=0\) for all flat \(R\)-modules \(F\). If \(R\in{\mathcal C}\) then \(M\) is cotorsion if and only if \(\text{Ext}_ R^ 1(E(R),M)=0\), \(E(R)\) being the injective envelope of \(R\). If \(R\in{\mathcal C}\) is indecomposable and \(F\) is a flat \(R\)-module then \(\text{Ext}^ 1_ R(E(R),F)\) is the completion of \(F\) with respect to the adic topology of all maximal ideals of \(R\). An extension of Harrison duality [\textit{D. K. Harrison}, Ann. Math., II. Ser. 69, 366-391 (1959; Zbl 0100.02901)] for injective torsion \(R\)-modules when \(R\in{\mathcal C}\) is given.
scientific article; zbMATH DE number 554251