Thesis (PhD (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: THE concept of a k-matrix in the full 2 2 matrix ring M2(R=hki), where R is an arbitrary unique
factorization domain (UFD) and k is an arbitrary nonzero nonunit in R, is introduced. We obtain
a concrete description of the centralizer of a k-matrix bB in M2(R=hki) as the sum of two subrings S1
and S2 ofM2(R=hki), where S1 is the image (under the natural epimorphism fromM2(R) toM2(R=hki))
of the centralizer in M2(R) of a pre-image of bB, and where the entries in S2 are intersections of certain
annihilators of elements arising from the entries of bB. Furthermore, necessary and sufficient conditions
are given for when S1 S2, for when S2 S1 and for when S1 = S2. It turns out that if R is a principal
ideal domain (PID), then every matrix in M2(R=hki) is a k-matrix for every k. However, this is not the
case in general if R is a UFD. Moreover, for every factor ring R=hki with zero divisors and every n > 3
there is a matrix for which the mentioned concrete description is not valid. Finally we provide a formula
for the number of elements of the centralizer of bB in case R is a UFD and R=hki is finite. / AFRIKAANSE OPSOMMING: DIE konsep van ’n k-matriks in die volledige 2 2 matriksring M2(R=hki), waar R ’n willekeurige
unieke faktoriseringsgebied (UFG) en k ’n willekeurige nie-nul nie-inverteerbare element in R
is, word bekendgestel. Ons verkry ’n konkrete beskrywing van die sentraliseerder van ’n k-matriks bB
in M2(R=hki) as die som van twee subringe S1 en S2 van M2(R=hki), waar S1 die beeld (onder die
natuurlike epimorfisme van M2(R) na M2(R=hki)) van die sentraliseerder in M2(R) van ’n trubeeld
vanbB is, en die inskrywings van S2 die deursnede van sekere annihileerders van elemente afkomstig van
die inskrywings van bB is. Verder word nodige en voldoende voorwaardes gegee vir wanneer S1 S2,
vir wanneer S2 S1 en vir wanneer S1 = S2. Dit blyk dat as R ’n hoofideaalgebied (HIG) is, dan is elke
matriks in M2(R=hki) ’n k-matriks vir elke k. Dit is egter nie in die algemeen waar indien R ’n UFG is
nie. Meer nog, vir elke faktorring R=hki met nuldelers en elke n > 3 is daar ’n matriks waarvoor die
bogenoemde konkrete beskrywing nie geldig is nie. Laastens word ’n formule vir die aantal elemente
van die sentraliseerder van bB verskaf, indien R ’n UFG en R=hki eindig is.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/5154 |
| Date | 12 1900 |
| Creators | Marais, Magdaleen Suzanne |
| Contributors | Van Wyk, L., University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Format | 87 p. |
| Rights | University of Stellenbosch |
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