A monoid-theoretical approach to infinite direct-sum decompositions of modules

Let $\mathcal C$ be a class of modules over a ring $R$, closed under direct sums over index sets of cardinality $\kappa$ 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 $\kappa$-indexed direct sums, and study the resulting $\kappa$-monoid $V^{\kappa}(\mathcal C)$. 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 $\lambda$ many elements, then all relations on $V^{\kappa}(\mathcal C)$ are induced by relations between direct sums indexed by sets of cardinality strictly less than $\lambda$. 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 $\kappa$-monoid $V^{\kappa}(R)$. Together with the realization result of Bergman and Dicks, this characterizes the $\kappa$-monoids appearing as $V^{\kappa}(R)$ for a hereditary ring. In general, the $\aleph_0$-monoid $V^{\aleph_0}(R)$ fully determines $V^{\kappa}(R)$. Herbera and P\v{r}\'ihoda's characterization of monoids of countably generated projective modules $V^*(R)$ over semilocal noetherian rings, yields a characterization of $V^{\kappa}(R)$ for these rings. We also characterize two-generated $\aleph_0$-monoids that appear as $V^{\aleph_0}(R)$ for hereditary rings $R$.