Reduction number one for finitely generated torsion-free modules

Let (R, m) be a two-dimensional regular local ring and let A be a finitely generated torsion-free R-module. If A is a complete module, then Dan Katz and Vijay Kodiyalam show A satisfies five conditions. They ask whether these five conditions are equivalent without assuming A to be complete. In a previous paper we determined all implications among these five conditions with one exception. In this paper, with an additional hypothesis on the units of R, we resolve the remaining case. Key words and phrases. integral closure, reduction of an ideal, reduction of a module, reduction number, Rees algebra.