The covering numbers of 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 $R$. If $R$ is a ring with unity, then we define the unital covering number ${\sigma }_u(R)$ to be the size of a minimal cover of $R$ by subrings that contain $1_R$ (if such a cover exists), and $R$ is ${\sigma }_u$-elementary if ${\sigma }_u(R)<{\sigma }_u(R/I)$ for every nonzero two-sided ideal of $R$. In this paper, we classify all $\sigma$-elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all ${\sigma }_u$-elementary rings and prove ${\sigma }_u(R)=\sigma (R)$ for every ${\sigma }_u$-elementary ring $R$. We also prove that, if $R$ is a ring without unity with a finite cover, then there exists a unital ring $R^{\prime }$ such that $\sigma (R)={\sigma }_u(R^{\prime })$, which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, if \[\mathscr{E}(N) := \{m : m \le N, σ(R) = m \text{ for some ring } R\},\] then we show that $|\mathcal{E}(N)|=\Theta (N/\log (N))$, which proves that almost all integers are not covering numbers of a ring.

arXiv commentThe original paper has been split in two: part of the material from the original version is contained in arXiv:2211.10313, and the rest is in this updated version