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Abstract
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A cover of a finite ring
is a
collection of proper subrings
of
such that
. If such a collection
exists, then
is called coverable, and the covering number of
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
be the field with
elements and let
be the ring of
upper triangular matrices
with entries from
.
We prove that if
,
then
has covering
number
, that
has covering number
4, and that when
is prime,
has
covering number
for all
.
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Keywords
covering number, upper triangular matrix ring, maximal
subring
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Mathematical Subject Classification 2010
Primary: 16P10
Secondary: 05E15
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Milestones
Received: 16 September 2018
Revised: 18 November 2018
Accepted: 5 March 2019
Published: 3 August 2019
Communicated by Scott T. Chapman
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