Representations over PID’s with three distinguished submodules

Abstract

Abstract. Let R be a principal ideal domain. The R-representations with one distinguished submodule are classified by a theorem of Gauß in the case of finite rank, and by the "Stacked Bases Theorem" of Cohen and Gluck in the case of infinite rank. Results of Hill and Megibben carry this classification even further. The R-representations with two distinguished pure submodules have recently been classified by Arnold and Dugas in the finite-rank case, and by the authors for countable rank. Although wild representation type prevails for R-representations with three distinguished pure submodules, an extensive category of such objects was recently classified by Arnold and Dugas. We carry their groundbreaking work further, simplifying the proofs of their main results and applying new machinery to study the structure of finite-and infinite-rank representations with two, three, and four distinguished submodules. We also apply these results to the classification of Butler groups, a class of torsion-free abelian groups that has been the focus of many investigations over the last sixteen years.