## Involve: A Journal of Mathematics

• Involve
• Volume 12, Number 6 (2019), 1005-1013.

### Covering numbers of upper triangular matrix rings over finite fields

#### Abstract

A cover of a finite ring $R$ is a collection of proper subrings ${S1,…,Sm}$ of $R$ such that $R= ⋃i=1mSi$. 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 $Fq$ be the field with $q$ elements and let $Tn(Fq)$ be the ring of $n×n$ upper triangular matrices with entries from $Fq$. We prove that if $q≠4$, then $T2(Fq)$ has covering number $q+1$, that $T2(F4)$ has covering number 4, and that when $p$ is prime, $Tn(Fp)$ has covering number $p+1$ for all $n≥2$.

#### Article information

Source
Involve, Volume 12, Number 6 (2019), 1005-1013.

Dates
Received: 16 September 2018
Revised: 18 November 2018
Accepted: 5 March 2019
First available in Project Euclid: 13 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.involve/1565661769

Digital Object Identifier
doi:10.2140/involve.2019.12.1005

Mathematical Reviews number (MathSciNet)
MR3990794

Zentralblatt MATH identifier
07116066

#### Citation

Cai, Merrick; Werner, Nicholas J. Covering numbers of upper triangular matrix rings over finite fields. Involve 12 (2019), no. 6, 1005--1013. doi:10.2140/involve.2019.12.1005. https://projecteuclid.org/euclid.involve/1565661769

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