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 {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 q4, 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.

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