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 are also of independent interest.

arXiv comment35 pages; final version

References in corpus (6)

• Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics
• Structural properties of subadditive families with applications to factorization theory
• Sets of Arithmetical Invariants in Transfer Krull Monoids
• Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules
• A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems
• On elasticities of locally finitely generated monoids

Cited by in corpus (3)

• An Abstract Factorization Theorem and Some Applications
• Factorization under Local Finiteness Conditions
• Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields