# Daniel Smertnig: A monoid-theoretical approach to infinite direct-sum decompositions

Let $C$ be a class of modules closed under isomorphisms and finite direct sums. The isomorphism classes together with the operation induced from the direct sum form a reduced commutative monoid $V(C)$. The arithmetic of $V(C)$ encodes the behavior of finite direct-sum decompositions in $C$. If $C$ is also closed under direct sums over index sets of some infinite cardinality $\kappa$, we obtain an additional infinitary operation on $V(C)$. We call the resulting structure a $\kappa$-monoid and investigate which $\kappa$-monoids appear as $\kappa$-monoids of modules. An extension property that we call the braiding property, and that is equivalent to a universal property in the category of $\kappa$-monoids plays a key role. Together with a classical theorem of Bergman and Dicks, it allows a description of the $\kappa$-monoids appearing over hereditary rings.

This talk is based on joint work with Z. Nazemian.

Vljudno vabljeni.

Roman Drnovšek in Primož Moravec