Lattices over Bass rings and graph agglomerations

Abstract

We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T (R). Results of Levy-Wiegand and Levy-Odenthal together with a study of the local case yield an explicit description of T (R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations-a natural class of monoids serving as combinatorial models for the factorization theory of T (R). As a consequence, the monoid T (R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T (R) and characterize when T (R) is half-factorial. (Factoriality, that is, torsionfree Krull-Remak-Schmidt-Azumaya, is characterized by a theorem of Levy-Odenthal.) The monoids of graph agglomerations introduced here may also be of independent interest to the factorization theory community.