#### Vol. 12, No. 6, 2019

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Covering numbers of upper triangular matrix rings over finite fields

### Merrick Cai and Nicholas J. Werner

Vol. 12 (2019), No. 6, 1005–1013
##### Abstract

A cover of a finite ring $R$ is a collection of proper subrings $\left\{{S}_{1},\dots ,{S}_{m}\right\}$ of $R$ such that $R={\bigcup }_{i=1}^{m}{S}_{i}$. If such a collection exists, then $R$ is called coverable, and the covering number of $R$ is the cardinality of the smallest possible cover. We investigate covering numbers for rings of upper triangular matrices with entries from a finite field. Let ${\mathbb{F}}_{q}$ be the field with $q$ elements and let ${T}_{n}\left({\mathbb{F}}_{q}\right)$ be the ring of $n×n$ upper triangular matrices with entries from ${\mathbb{F}}_{q}$. We prove that if $q\ne 4$, then ${T}_{2}\left({\mathbb{F}}_{q}\right)$ has covering number $q+1$, that ${T}_{2}\left({\mathbb{F}}_{4}\right)$ has covering number 4, and that when $p$ is prime, ${T}_{n}\left({\mathbb{F}}_{p}\right)$ has covering number $p+1$ for all $n\ge 2$.

##### Keywords
covering number, upper triangular matrix ring, maximal subring
Primary: 16P10
Secondary: 05E15