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, torsion-free 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.