On Representing Matroids via Modular Independence
Koji Imamura, Keisuke Shiromoto
Abstract
We study a matrix-based notion of matroid representation over local commutative rings obtained by replacing linear independence with modular independence. This construction always defines an independence system, though not necessarily a matroid. Under a mild nilpotent hypothesis, we show that chain rings are exactly the local rings for which the minimal number of generators is monotone on finitely generated submodules, and over commutative chain rings we obtain a criterion for the associated independence system to be a matroid. For codes over finite commutative chain rings, we identify puncturing with deletion, show that shortening agrees with contraction under a contractibility hypothesis, and establish duality for free codes. We further derive bounds for simple and uniform matroids, prove that the uniform matroid $U_{2, n}$ is representable if and only if the size $n$ is at most the sum of the cardinalities of the local ring and its unique maximal ideal, and show that all excluded minors for $\mathbb{F}_{4}$-representability are representable over $\mathbb{Z}/4\mathbb{Z}$. The examples also include ring representations of matroids not representable over any field, such as the Vámos matroid over $\mathbb{Z}/8\mathbb{Z}$.