Direct Sum Decompositions Over Two-Dimensional Local Domains

Abstract

Given a ring R and a class 𝒞 of R-modules, one can ask whether or not every element of 𝒞 decomposes uniquely as a direct sum of indecomposable elements of 𝒞. If not, one can further ask if it is possible for an element of 𝒞 to decompose both as the direct sum of s indecomposable elements of 𝒞 and as the direct sum of t indecomposable elements of 𝒞 where s ≠ t. In this article, we investigate these questions when R is a two-dimensional analytically normal domain and 𝒞 is the class of finitely generated torsion-free R-modules.