Covering numbers of upper triangular matrix rings over finite fields

Cai, Merrick; Werner, Nicholas J.

doi:10.2140/involve.2019.12.1005

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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

DOI: 10.2140/involve.2019.12.1005

Abstract

A cover of a finite ring $R$ is a collection of proper subrings ${S_{1}, \dots, S_{m}}$ of $R$ such that $R = ⋃_{i = 1}^{m} S_{i}$ . 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 $F_{q}$ be the field with $q$ elements and let $T_{n} (F_{q})$ be the ring of $n \times n$ upper triangular matrices with entries from $F_{q}$ . We prove that if $q \neq 4$ , then $T_{2} (F_{q})$ has covering number $q + 1$ , that $T_{2} (F_{4})$ has covering number 4, and that when $p$ is prime, $T_{n} (F_{p})$ has covering number $p + 1$ for all $n \geq 2$ .

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Keywords

covering number, upper triangular matrix ring, maximal subring

Mathematical Subject Classification 2010

Primary: 16P10

Secondary: 05E15

Milestones

Received: 16 September 2018

Revised: 18 November 2018

Accepted: 5 March 2019

Published: 3 August 2019

Communicated by Scott T. Chapman

Authors

Merrick Cai
Kings Park High School Kings Park, NY United States

Nicholas J. Werner
Department of Mathematics, Computer and Information Science SUNY College at Old Westbury Old Westbury, NY United States

Vol. 12, No. 6, 2019