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

Eric Swartz; Nicholas J. Werner

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

Rings and Algebras 2022-11-21 v1 Combinatorics

Authors: Eric Swartz , Nicholas J. Werner

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.

Keywords

associative rings ring covers covering number sigma-elementary rings Jacobson radical noncommutative quotient

Cite

@article{arxiv.2211.10313,
  title  = {A new infinite family of $\sigma$-elementary rings},
  author = {Eric Swartz and Nicholas J. Werner},
  journal= {arXiv preprint arXiv:2211.10313},
  year   = {2022}
}

Comments

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

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

Abstract

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