  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 $\sigma(R)$ 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 show that if $R$ has a finite covering number, then the calculation
of $\sigma(R)$ 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 $\sigma(R)=\sigma(R/J)$,
or $\sigma(R)=p^d+1$ for some $d \geqslant 1$. As a byproduct, we classify all
commutative $\sigma$-elementary rings with a finite covering number and
characterize the integers that occur as the covering number of a commutative
ring.
