Lattices over Bass rings and graph agglomerations

TL;DR

This paper explores the factorization properties of modules over Bass rings by linking their monoids to graph agglomerations, revealing finite-type transfer Krull structures and conditions for half-factoriality.

Contribution

It introduces a transfer homomorphism from the monoid of modules over Bass rings to graph agglomeration monoids, enabling new finiteness and factorization results.

Findings

The monoid T(R) is transfer Krull of finite type.

Conditions for T(R) to be half-factorial are characterized.

Graph agglomeration monoids are of independent mathematical interest.

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 are also of independent interest.