Covering numbers of upper triangular matrix rings over finite fields

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 $n\times n$ 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 $n\ge 2$.