Vector spaces over finite commutative rings

TL;DR

This paper generalizes the theory of vector spaces from finite fields to finite commutative rings, deriving formulas for subspaces and exploring applications to arcs and caps.

Contribution

It introduces new formulas for subspaces over commutative rings and extends classical concepts like arcs and caps to this broader setting.

Findings

Derived Anzahl formulas for subspaces over commutative rings

Established dimensional formulas for subspaces in this context

Applied these results to study arcs and caps

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.