Covering numbers of commutative rings

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 $σ(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $σ$-elementary if $σ(R) < σ(R/I)$ 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 $σ(R)$ 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 $σ(R)=σ(R/J)$, or $σ(R)=p^d+1$ for some $d \geqslant 1$. As a byproduct, we classify all commutative $σ$-elementary rings with a finite covering number and characterize the integers that occur as the covering number of a commutative ring.

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