Direct-sum behavior of modules over one-dimensional rings

Let R be a reduced, one-dimensional Noetherian local ring whose integral closure $$\overline{R}$$ is finitely generated over R. Since $$\overline{R}$$ is a direct product of finitely many principal ideal domains (one for each minimal prime ideal of R), the indecomposable finitely generated $$\overline{R}$$ -modules are easily described, and every finitely generated $$\overline{R}$$ -module is uniquely a direct sum of indecomposable modules. In this article we will see how little of this good behavior trickles down to R. Indeed, there are relatively few situations where one can describe all of the indecomposable R-modules, or even the torsion-free ones. Moreover, a given finitely generated module can have many different representations as a direct sum of indecomposable modules.