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Let $R$ be a ring and let $\mathcal C$ be a small class of right $R$-moduleswhich is closed under finite direct sums, direct summands, and isomorphisms.Let $\mathcal V (\mathcal C)$ denote a set of representatives of isomorphismclasses in $\mathcal C$ and, for any module $M$ in $\mathcal C$, let $[M]$denote the unique element in $\mathcal V (\mathcal C)$ isomorphic to $M$. Then$\mathcal V (\mathcal C)$ is a reduced commutative semigroup with operationdefined by $[M] + [N] = [M \oplus N]$, and this semigroup carries allinformation about direct-sum decompositions of modules in $\mathcal C$. Thissemigroup-theoretical point of view has been prevalent in the theory ofdirect-sum decompositions since it was shown that if $\operatorname{End}_R (M)$is semilocal for all $M\in \mathcal C$, then $\mathcal V (\mathcal C)$ is aKrull monoid. Suppose that the monoid $\mathcal V (\mathcal C)$ is Krull with afinitely generated class group (for example, when $\mathcal C$ is the class offinitely generated torsion-free modules and $R$ is a one-dimensional reducedNoetherian local ring). In this case we study the arithmetic of $\mathcal V(\mathcal C)$ using new methods from zero-sum theory. Furthermore, based onmodule-theoretic work of Lam, Levy, Robson, and others we study the algebraicand arithmetic structure of the monoid $\mathcal V (\mathcal C)$ for certainclasses of modules over Pr\"ufer rings and hereditary Noetherian prime rings.

A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems