# Journal of the European Mathematical Society

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**Volume 18, Issue 4, 2016, pp. 681–731**

**DOI: 10.4171/JEMS/600**

Published online: 2016-03-08

Matroids over a ring

Alex Fink^{[1]}and Luca Moci (1) Queen Mary University of London, UK

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.

*Keywords: *Matroid, module over Dedekind ring, arithmetic matroid, valuated matroid, arithmetic Tutte polynomial, tropical flag Dressian, Tutte–Grothendieck ring

Fink Alex, Moci Luca: Matroids over a ring. *J. Eur. Math. Soc.* 18 (2016), 681-731. doi: 10.4171/JEMS/600