A new infinite family of $\sigma$-elementary rings

TL;DR

This paper introduces a new infinite family of rings called $\sigma$-elementary rings, characterized by minimal covers, and provides the first examples with nontrivial Jacobson radical and noncommutative quotients.

Contribution

It presents the first examples of $\sigma$-elementary rings with nontrivial Jacobson radical and noncommutative quotients, expanding understanding of ring coverings.

Findings

Constructed the first examples of such rings.

Determined the covering numbers of these rings.

Analyzed properties related to Jacobson radical and noncommutativity.

Abstract

A cover of an associative (not necessarily commutative nor unital) 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 provide the first examples of $\sigma$-elementary rings $R$ that have nontrivial Jacobson radical $J$ with $R/J$ noncommutative, and we determine the covering numbers of these rings.