In this article we consider finitely generated torsion-free modules over certain one-dimensional commutative Noetherian rings R. We assume there exists a positive integer N R such that, for every indecomposable R-module M and for every minimal prime ideal P of R, the dimension of M P , as a vector space over the field R P , is less than or equal to N R . If a nonzero indecomposable R-module M is such that all the localizations M P as vector spaces over the fields R P have the same dimension r , for every minimal prime P of R, then r = 1, 2, 3, 4 or 6. Let n be an integer ≥ 8. We show that if M is an R-module such that the vector space dimensions of the M P are between n and 2n − 8, then M decomposes non-trivially. For each n ≥ 8, we exhibit a semilocal ring and an indecomposable module for which the relevant dimensions range from n to 2n−7. These results require a mild equicharacteristic assumption; we also discuss bounds in the non-equicharacteristic case. MSC: Primary: 13C05; secondary: 13E05; 13G05
IntroductionOver the past forty years, a great deal of progress has been made towards determining representation type over certain Noetherian one-dimensional rings, that is, on the question: What are the isomorphism classes of the indecomposable modules? In the classical work from the sixties, the rings studied are one-dimensional domains that are finitely generated over the integers and are contained in algebraic number fields. See, for example [11,12,16,18,19].In this article we consider a larger class of one-dimensional rings for which, over the past twenty years, a concept of the size or rank of a module, as well as reasonable criteria so that the indecomposable modules have finite rank, have been developed [1,3,4,6,13,15,20,21]. We study the question: For those rings such that a bound exists on the ranks of indecomposable modules, what are the bounds?In order to describe the setting and the results of this paper, we start with some terminology. We also give some history, recent developments, and useful previous results.