  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.
