# Contributions to centralizers in matrix rings

 dc.contributor.advisor Van Wyk, L. dc.contributor.author Marais, Magdaleen Suzanne dc.contributor.other University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. dc.date.accessioned 2010-11-19T14:12:32Z dc.date.accessioned 2010-12-15T10:16:00Z dc.date.available 2010-11-19T14:12:32Z dc.date.available 2010-12-15T10:16:00Z dc.date.issued 2010-12 dc.identifier.uri http://hdl.handle.net/10019.1/5154 dc.description Thesis (PhD (Mathematics))--University of Stellenbosch, 2010. dc.description.abstract ENGLISH ABSTRACT: THE concept of a k-matrix in the full 2 2 matrix ring M2(R=hki), where R is an arbitrary unique en 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. dc.description.abstract AFRIKAANSE OPSOMMING: DIE konsep van ’n k-matriks in die volledige 2 2 matriksring M2(R=hki), waar R ’n willekeurige af 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. dc.format.extent 87 p. dc.language.iso en dc.publisher Stellenbosch : University of Stellenbosch dc.subject Centralizers dc.subject Matrix rings dc.subject Dissertations -- Mathematics en dc.subject Theses -- Mathematics en dc.subject Matrices en dc.title Contributions to centralizers in matrix rings en dc.rights.holder University of Stellenbosch
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