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

 Download this article For screen For printing  Recent Issues  The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals  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