Covering numbers of commutative rings

TL;DR

This paper investigates the minimal number of proper subrings needed to cover a unital ring, especially focusing on finite, commutative, and Jacobson radical conditions, providing classifications and characterizations of such covering numbers.

Contribution

It reduces the calculation of covering numbers to finite rings of characteristic p with Jacobson radical of nilpotency 2 and classifies commutative sigma-elementary rings with finite covering numbers.

Findings

If R has finite covering number, it can be reduced to a finite ring of characteristic p with J^2=0.

For rings with commutative R/J, the covering number is either equal to that of R/J or p^d+1.

The paper classifies all commutative sigma-elementary rings with finite covering number.

Abstract

A cover of a unital, associative (not necessarily commutative) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $\sigma$-elementary if $\sigma(R) < \sigma(R/I)$ for every nonzero two-sided ideal $I$ of $R$. In this paper, we show that if $R$ has a finite covering number, then the calculation of $\sigma(R)$ can be reduced to the case where $R$ is a finite ring of characteristic $p$ and the Jacobson radical $J$ of $R$ has nilpotency 2. Our main result is that if $R$ has a finite covering number and $R/J$ is commutative (even if $R$ itself is not), then either $\sigma(R)=\sigma(R/J)$, or $\sigma(R)=p^d+1$ for some $d \geqslant 1$. As a byproduct, we classify all commutative $\sigma$-elementary rings with a finite covering number and characterize the integers that occur as the covering number of a commutative ring.