Categorial properties of compressed zero-divisor graphs of finite commutative rings

TL;DR

This paper introduces a new compressed zero-divisor graph for finite commutative rings, demonstrating its categorical properties and using it to characterize key classes of rings.

Contribution

It defines the compressed zero-divisor graph via associatedness, proves it induces a functor that preserves categorical products, and applies it to classify certain rings.

Findings

The graph is the optimal compression inducing a functor.

The functor preserves categorical products.

It characterizes local and principal ideal rings.

Abstract

We define a compressed zero-divisor graph $\varTheta(K)$ of a finite commutative unital ring $K$, where the compression is performed by means of the associatedness relation. We prove that this is the best possible compression which induces a functor $\varTheta$, and that this functor preserves categorial products (in both directions). We use the structure of $\varTheta(K)$ to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.