Involve: A Journal of Mathematics

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

Covering numbers of upper triangular matrix rings over finite fields

Merrick Cai and Nicholas J. Werner

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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 n2.

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

Subjects
Primary: 16P10: Finite rings and finite-dimensional algebras {For semisimple, see 16K20; for commutative, see 11Txx, 13Mxx}
Secondary: 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]

Keywords
covering number upper triangular matrix ring maximal subring

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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