On rings as unions of four subrings

TL;DR

This paper classifies unital rings with a covering number of four and provides a method to determine the minimal number of proper subrings needed to cover finite rings, including finite local rings with prime residue fields.

Contribution

It introduces a classification strategy for unital rings based on their covering number and specifically characterizes rings with covering number four.

Findings

Classified unital rings with covering number four.

Computed covering numbers for finite local rings with prime residue fields.

Developed a general strategy for classifying rings by covering number.

Abstract

The covering number of an associative ring $R$ is the minimal number of proper subrings whose union is $R$. We establish a strategy to classify unital rings of a given finite covering number, and obtain a classification of unital rings whose covering number is four. Along the way we compute the covering number of every finite local ring whose residue field has prime order.