# Matroids over a ring

Abstract : 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, respectively. More generally, whenever $R$ is a Dedekind domain, we extend the usual properties and operations holding for matroids (e.g., duality), and we compute the Tutte-Grothendieck group of matroids over $R$.
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Conference papers

Cited literature [13 references]

https://hal.inria.fr/hal-01229708
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dmAS0114.pdf
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• HAL Id : hal-01229708, version 1

### Citation

Alex Fink, Luca Moci. Matroids over a ring. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.157-168. ⟨hal-01229708⟩

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