论文标题
覆盖通勤戒指的数量
Covering numbers of commutative rings
论文作者
论文摘要
$ r $的合适子环的集合，其套件的封面是$ r $的封面（不一定是可交换）环$ r $。如果存在这样的盖子，则$ r $的覆盖号$σ（r）$是最小盖子的基础性，如果对于每个非零的两边$ i $ $ r $ $ $ r $ $ $ r $ $σ（r）<σ（r）<σ（r）<σ（r）<σ（r/i）$，则称为$σ$ - elementary。在本文中，我们表明，如果$ r $具有有限的覆盖号码，那么可以将$σ（r）$的计算减少到那些情况下，$ r $是有限的特征性$ p $的有限戒指，而雅各布森自由基$ j $ $ r $的$ r $具有nilpotency 2。 $σ（r）=σ（r/j）$，或$σ（r）= p^d+1 $，对于某些$ d \ geqslant 1 $。作为副产品，我们将所有交换性$σ$ elementary环与有限的覆盖号进行分类，并将出现的整数表征为交换环的覆盖次数。
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.
