Classification of Modules with Two Distinguished Pure Submodules and Bounded Quotients

In this paper, we consider modules over principal ideal domains R. The objects are free R- modules F with two distinguished pure submodules F 0 and F1 with F 0 ∩ F1 = 0 and bounded quotient F/(F 0 ⊕ F 1) and morphisms are the usual R-homomorphisms which preserve the distinguished submodules. This category is denoted by cRep2.R and its objects, we say the cR2-modules are denoted by F = (F, F0, F 1). The rank of a cR2-module F is the rank of the free R-module F. We will show that cR2 -@#@ modules are direct sums of indecomposable cR2-modules of rank 1 or 2. The infinite series of indecomposable cR2-modules is well-known and given explicitly after our Main Theorem 1.4. The result was first shown for cR2modules of finite rank in Arnold and Dugas [4], then for countable rank, using heavy machinery due to Hill and Megibben [25] in Files and Göbel [20]. Our proof for arbitrary rank is based on [20] and illustrates the importance of Hill’s notion of an axiom-3 family of modules. The Main Theorem is applied to a classification of Butler groups with two critical types. 1 2