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 = \mathbb 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 \mathbb Z can be obtained as an evaluation of the class of the matroid in the Tutte–Grothendieck ring.