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.
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Smertnig would like to thank the Department for the hospitality received.
Funding
Open access funding provided by University of Graz. Smertnig was supported by the Austrian Science Fund (FWF) project J4079-N32. Part of the research was conducted while Smertnig was visiting the University of Waterloo.
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Baeth, N.R., Smertnig, D. Lattices over Bass Rings and Graph Agglomerations. Algebr Represent Theor 25, 669–704 (2022). https://doi.org/10.1007/s10468-021-10040-2
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DOI: https://doi.org/10.1007/s10468-021-10040-2