Vector spaces over finite commutative rings

TL;DR

This work extends the classical theory of subspace orbits and dimension formulas from finite fields to vector spaces over finite commutative rings, introducing and exploiting the McCoy rank $\mathrm{rk}(A)$ and unimodular bases. A hierarchical matrix-method framework is developed: first for local rings via the residue map $\pi$, then for products of local rings using componentwise projections, to obtain enumeration formulas for matrices of fixed McCoy rank, sizes of subspace families, and canonical representations under $GL_n(R)$. The paper further defines singular linear spaces and attenuated spaces, deriving orbit-counting formulas in that setting, and applies the theory to arcs and caps, showing that their maximal sizes over rings are governed by the minimum across component residue fields. Concrete corollaries for small dimensions and links to known results over fields and residue rings are provided, alongside avenues for future work on $r$-intersecting families and related combinatorial structures. Overall, the results unify and extend subspace enumeration, dimension formulas, and geometric configurations from fields to the broader regime of finite commutative rings.

Abstract

Vector spaces over finite fields and Anzahl formulas of subspaces were studied by Wan (Geometry of Classical Groups over Finite Fields, Science Press, 2002). As a generalization, we study vector spaces and singular linear spaces over commutative rings, and obtain some Anzahl formulas and dimensional formula for subspaces. Moreover, we discuss arcs and caps by using these subspaces.

Theorems & Definitions (31)

• Proposition 1.1
• Lemma 2.1
• Corollary 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Corollary 2.7
• Lemma 3.1
• Lemma 3.2