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

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.

The content of the earlier paper "The covering numbers of rings" (arXiv:2112.01667) has been split into two parts. This article contains one part, and the updated version of arXiv:2112.01667 contains the rest