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# Matrix Algebras: Equivalent Ring Relations and Special Presentations

 dc.contributor.advisor Agnarsson, Geir dc.contributor.author Mendelson, Samuel Stephen dc.creator Mendelson, Samuel Stephen dc.date.accessioned 2018-10-22T01:19:47Z dc.date.available 2018-10-22T01:19:47Z dc.date.issued 2017 dc.identifier.uri https://hdl.handle.net/1920/11246 dc.description.abstract Recognizing when a ring is a matrix ring is of significant importance in the study of algebra. A well-known result in noncommutative ring theory states that a ring $R$ is a matrix ring if and only if it contains a set of $n\times n$ matrix units $\{e_{ij}\}_{i,j=1}^n$; in which case $R\cong M_2(S)$ for some $S$ that can be completely described in terms of these matrix units. However, finding and verifying a set of matrix units can be difficult. A more recent result states that a ring $R$ is an $(m+n)\times(m+n)$ matrix ring if, and only if, it contains three elements, $a$, $b$, and $f$, satisfying the two relations $af^m+f^nb=1$ and $f^{m+n}=0$, in which case $R\cong M_{m+n}(S)$ for some $S$. Under these relations very little is known about the structure of $S$. In this dissertation we investigate algebras over a commutative ring $A$ (or a field $k$) with elements $x$ and $y$ that satisfy the relations $x^iy+yx^j=1$ and $y^2=0$. We develop results about the structure of these algebras and their underlying rings when $\gcd(i,j)=1$ and then generalize these results for all $i$ and $j$. We then present some interesting examples demonstrating the more subtle characteristics of these algebras. Finally, we develop techniques to see when these algebras can be mapped to $2\times 2$ matrix rings. dc.format.extent 83 pages dc.language.iso en dc.rights Copyright 2017 Samuel Stephen Mendelson dc.subject Mathematics en_US dc.subject Diamond Lemma en_US dc.subject Free Algebras en_US dc.subject Matrix Recognition en_US dc.subject Matrix Relations en_US dc.subject Noncommutative Algebra en_US dc.title Matrix Algebras: Equivalent Ring Relations and Special Presentations dc.type Dissertation thesis.degree.level Ph.D. thesis.degree.discipline Mathematics thesis.degree.grantor George Mason University
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