Noetherian stable domains
\textit{R. B. Warfield} jun. [Math. Z. 107, 189-200 (1968; Zbl 0169.03602)] showed that for torsion-free abelian groups \(A\) and \(B\), when \(A\) has rank 1, the natural map \(\Hom(A,B)\otimes_{\text{End}(A)} A \to B\) is an embedding with image equal to the subgroup of \(B\) whose types are at least of type \(A\). The goal of the paper under review is to determine the extent to which properties of the integers come into play in obtaining Warfield's duality result by investigating noetherian domains which support Warfield's result. For the purpose of this study, the author introduces the notions of solvable and weakly solvable integral domains: An integral domain \(R\) is called solvable if for any rank 1 module \(A\), the image of the natural map \(\Hom(A,B)\otimes_{\text{End}_R(A)}A \to B\) contains the elements whose types, in the sense of \textit{J. D. Reid} [in: Abelian groups and noncommutative rings, Collect. Pap. in Mem. of R. B. Warfield, Contemp. Math. 130, 361-370 (1992; Zbl 0782.20045)] are at least of type \(A\), whenever \(B\) is a torsion-free \(\text{End}_R(A)\)-module. Also, \(R\) is called weakly solvable if for every rank 1 module \(A\), the natural map \(\Hom(A,B)\otimes_{\text{End}_R(A)}A \to B\) is an embedding for any torsion-free \(\text{End}_R(A)\)-module. Then, the author investigates the relationship of these two notions with those of stable ring (i.e. a noetherian ring such that every ideal is projective over its endomorphism ring) introduced by \textit{J. D. Sally} and \textit{W. V. Vasconcelos} [J. Pure Appl. Algebra 4, 319-336 (1974; Zbl 0284.13010)], and shows that for a noetherian domain the three notions are equivalent. The author also shows several results involving stable domains, and in particular that an extension of a stable domain in its quotient field is stable. The end of the paper studies the relationship of the previous notions with those of Warfield domain (i.e. an integral domain such that for any rank 1 module \(A\), the torsion-free modules \(B\) of finite rank satisfying the natural isomorphism \(\Hom_R(\Hom_R(B,A),A) \cong B\) are precisely the modules which embed as \(\text{End}_R(A)\)-modules of \(A^n\) for some \(n\)) introduced by \textit{S. Bazzoni} and \textit{L. Salce} [J. Algebra 185, No. 3, 836-868 (1996; Zbl 0873.13020)]. The author shows that a noetherian Warfield domain is solvable, and that the converse is untrue in general. Furthermore, he shows that if \(R\) is a noetherian domain such that its integral closure in its quotient field is a finitely generated \(R\)-module, then the four notions (stable, solvable, weakly solvable, Warfield domain) are equivalent and characterised by the fact that every ideal of \(R\) is two-generated.
scientific article; zbMATH DE number 1203899
scientific article