Authors: Alex Fink ^1,2; Luca Moci ³

• 1 University of North Carolina [Charlotte] [MSRI]
• 2 Mathematical Sciences Research Institute [MSRI]
• 3 Institut de Mathématiques de Jussieu - Paris Rive Gauche

We introduce the notion of a matroid $M$ over a commutative ring $R$, assigning to every subset of the ground set an $R$-module according to some axioms. When $R$ is a field, we recover matroids. When $R=\mathbb{Z}$, and when $R$ is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, respectively. More generally, whenever $R$ is a Dedekind domain, we extend the usual properties and operations holding for matroids (e.g., duality), and we compute the Tutte-Grothendieck group of matroids over $R$.