Hereditary Noetherian prime rings. I: Integrality and simple modules
This paper is the first of a trilogy with the aim of raising the structure theory of projective modules over a Noetherian hereditary prime (HNP) ring \(R\) to a level roughly equivalent to what has been achieved for the classical case of a (commutative) Dedekind domain. Part~II [J. Algebra~218, No. 2, 338-372 (1999)] is reviewed below (see Zbl 0946.16017); Part~III has appeared [J. Algebra~225, No. 1, 275-298 (2000; Zbl 0955.16025)]. The first part of this series lays the foundation of the theory by exploring the relationship between \(R\) and its Dedekind closures which play a role similar to that of maximal orders in the theory of classical orders over a Dedekind domain. The notion of a Dedekind closure of \(R\) is due to \textit{T. J. Hodges} [Trans. Am. Math. Soc.~ 302, No. 2, 751-767 (1987; Zbl 0634.16011)] who characterized these objects as the minimal Dedekind prime overrings of \(R\) (where ``overring'' means a ring between \(R\) and its Goldie quotient ring). He also showed that each one of them is a directed union of overrings that are finitely generated as right \(R\)-modules. In the paper under review, Dedekind closures are characterized as Dedekind prime right integral overrings, where an overring \(S\) of \(R\) is called right integral over \(R\) if any subring generated by \(R\) together with finitely many elements of \(S\) is finitely generated as a right \(R\)-module. Furthermore, Dedekind closures are shown to coincide with the maximal right integral overrings of \(R\). The authors provide a precise description the simple modules over a Dedekind closure in terms of certain sets of simple \(R\)-modules, called towers, and thus get a grip on factors of finite length of projective \(R\)-modules, which is vital for the later parts of this project.
scientific article; zbMATH DE number 1348162
scientific article