Speaker: Nick Werner (SUNY Old Westbury)
Title: The Covering Number of Rings
Abstract: A cover of a group $G$ is a collection of proper subgroups of $G$ whose union equals $G$. If such a cover exists, then the covering number $\sigma(G)$ of $G$ is the cardinality of a minimal cover. Studies on covering numbers of groups and the integers that occur as such covering numbers have a long history, and date back at least to work by Scorza in 1926. Despite extensive study, there are still many open problems relating to covering numbers of groups, and research on this topic is ongoing.
In this talk, we will discuss the analogous notions of covers and covering numbers for rings. As with groups, a cover of a ring $R$ is a collection of proper subrings of $R$ whose union is all of $R$, and the covering number $\sigma(R)$ is the size of a minimal cover (if one exists). Covering problems for rings began to receive attention around 2010, and have generally been more tractable than those for groups. We will summarize the known results on this topic, and will present some recent theorems of Eric Swartz and the speaker that can be used to determine $\sigma(R)$ for any ring $R$ with a finite covering number. As a corollary, we show that there are infinitely many positive integers that are not the covering number of a ring; in fact, almost all integers are not covering numbers of rings. These results are obtained by first reducing the general problem to the case of finite rings of characteristic $p$ whose Jacobson radical is 2-nilpotent, and then focusing on four families of rings where formulas for $\sigma(R)$ are either known or can be explicitly derived.