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.

中文翻译：

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覆盖交换环数

一个单位的、结合的（不一定是可交换的）环 $R$ 的覆盖是 $R$ 的适当子环的集合，其集合论并集等于 $R$。如果存在这样的覆盖，则 $R$ 的覆盖数 $\sigma(R)$ 是最小覆盖的基数，如果 $\sigma(R) < \sigma(R/I)$ 对于 $R$ 的每个非零两侧理想 $I$。在本文中，我们证明如果 $R$ 具有有限覆盖数，则 $\sigma(R)$ 的计算可以简化为 $R$ 是特征 $p$ 的有限环和 Jacobson $R$ 的部首 $J$ 具有幂零性 2。我们的主要结果是，如果 $R$ 具有有限覆盖数并且 $R/J$ 是可交换的（即使 $R$ 本身不是），则 $\sigma (R)=\sigma(R/J)$, 或 $\sigma(R)=p^d+1$ 对于某些 $d \geqslant 1$。作为副产品，