2024/01/16 by Zahra Nazemian, Nazemian, Zahra, Daniel Smertnig +1
Computer Science · Mathematics · #20M13 #20M75 #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Primary 16D70 #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #Secondary 16D40
paper · pdf · doi:10.48550/arxiv.2401.08203
openalex publication_date 2024/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathcal C be a class of modules over a ring R, closed under direct sums over index sets of cardinality κ and isomorphisms, and such that the isomorphism classes form a set. The monoid of modules V(\mathcal C) encodes the behavior of finite direct-sum decompositions of modules in \mathcal C. We endow V(\mathcal C) with an additional operation reflecting κ-indexed direct sums, and study the resulting κ-monoid Vκ(\mathcal C). The braiding-property and an equivalent universal property, allow us to show: if every module in \mathcal C is a direct sum of modules generated by strictly fewer than λ many elements, then all relations on Vκ(\mathcal C) are induced by relations between direct sums indexed by sets of cardinality strictly less than λ. A theorem of Kaplansky states that every projective module is a direct sum of countably generated modules. We augment this, showing that also all relations between infinite direct sums of projective modules are induced from those between countable direct sums of countably generated projective modules. If every projective module over a ring R is a direct sum of finitely generated projective modules, then the monoid of finitely generated projective modules V(R) completely determines the κ-monoid Vκ(R). Together with the realization result of Bergman and Dicks, this characterizes the κ-monoids appearing as Vκ(R) for a hereditary ring. In general, the ℵ0-monoid Vℵ0(R) fully determines Vκ(R). Herbera and Příhoda's characterization of monoids of countably generated projective modules V^*(R) over semilocal noetherian rings, yields a characterization of Vκ(R) for these rings. We also characterize two-generated ℵ0-monoids that appear as Vℵ0(R) for hereditary rings R.