Covering numbers of rings

Abstract: A cover of a ring $R$ is a collection $C$ of proper subrings of $R$ such that $R = \bigcup_{S \in C} S$. If such a collection exists, then $R$ is called coverable, and the covering number of $R$ is the cardinality of the smallest possible cover. Questions that have been considered on this topic include determining covering numbers for certain families of rings, or classifying all rings with a given covering number. As we will demonstrate, many of these questions can be reduced to the case of finite rings of characteristic $p$.

The analogous problem of finding covering numbers of groups has been extensively studied. While there are parallels between the group setting and the ring setting, much less is known in the case of rings. We will survey the known results on covering numbers of rings, and mention some conjectures and open problems, among them the unresolved question of whether there exists a ring with covering number 13.

combinatoricscomplex variablesfunctional analysisgeneral mathematicsgroup theoryK-theory and homologynumber theoryoperator algebrasquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

( slides )

Comments: Password to access the talk is the order of the symmetric group $S_9$.

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GAG seminar

Series comments: Description: Non-specialized research seminar

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