The minimal number of generators of an invertible ideal

Abstract

All rings in this paper are commutative with unity; we will deal mainly with integral domains. Let R be a ring with total quotient ring K. A fractional ideal I of R is invertible if II−1 = R; equivalently, I is a projective module of rank 1 (see, e.g., [Eis95, Section 11.3]). Here, I−1 = (R : I) = {x ∈ K |xI ⊆ R}. Moreover, a projective R-module of rank 1 is isomorphic to an invertible ideal. (We use the term “ideal” in the sense of an integral ideal). We denote the minimal number of generators of an ideal I of R by νR(I). If R is a Dedekind domain, equivalently, a domain in which each nonzero ideal is invertible, then νR(I) ≤ 2 for each nonzero ideal I; moreover, I is strongly 2generated, in the sense that one of the generators can be an arbitrary nonzero element of I. A Dedekind domain is characterized as an integrally closed Noetherian domain of Krull dimension 1. It turns out that of these three properties, Krull dimension 1 always implies that an invertible ideal is 2generated, as was shown by Sally and Vasconcelos in [SV74]. R. Heitmann generalized this fact to arbitrary finite Krull dimension: an invertible ideal of an n-dimensional domain R is strongly n + 1-generated (see Section 3). Moreover, this result is sharp, in the sense that for each n ≥ 1 there exists an n-dimensional domain R, even Prufer, with an invertible ideal requiring n+ 1 generators: see the examples in Sections 3, 4 and 5. The general problem of determining the minimal number of generators for an invertible ideal of a domain was first studied by Gilmer and Heinzer in [GH70]. Among other fundamental results, they provide sufficient conditions for an invertible ideal to have the property that it can be generated by two elements. The question whether a finitely generated ideal of a Prufer domain can be always be generated by 2 elements was first raised by Gilmer around 1964 [Swa84]. Recall that a Prufer domain is an integral domain in which each