Matroids over a ring

Understand

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.