Matroids Over a Ring

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 = ℤ, and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids i.e. tropical linear spaces, respectively.More generally, whenever R is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and we explicitly describe the structure of the matroids over R. Furthermore, we compute the Tutte-Grothendieck ring of matroids over R. We also show that the Tutte quasi-polynomial of a matroid over ℤ can be obtained as an evaluation of the class of the matroid in the Tutte-Grothendieck ring.