{"id":{"repo_id":"gmu","oai_identifier":"oai:MARS:1920/11246"},"canonical_url":"https://search.dev.ndltd.org/etd/gmu/oai:MARS:1920/11246","repository":{"repo_id":"gmu","name":"George Mason University","base_url":"https://mars.gmu.edu/server/oai/request"},"display":{"title":"Matrix Algebras: Equivalent Ring Relations and Special Presentations","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.","abstract_html":"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 <span class=\"etd-inline-math\">\\{e<sub>ij</sub>\\}<sub>i,j=1</sub><sup>n</sup></span>; in which case <span class=\"etd-inline-math\">R\\cong M<sub>2</sub>(S)</span> 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 <span class=\"etd-inline-math\">af<sup>m</sup>+f<sup>n</sup>b=1</span> and <span class=\"etd-inline-math\">f<sup>m+n</sup>=0</span>, in which case <span class=\"etd-inline-math\">R\\cong M<sub>m+n</sub>(S)</span> 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 <span class=\"etd-inline-math\">x<sup>i</sup>y+yx<sup>j</sup>=1</span> and <span class=\"etd-inline-math\">y<sup>2</sup>=0</span>. 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.","abstract_has_math":true,"creators":["Mendelson, Samuel Stephen"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017","date_published":"2017","updated_at":"2026-07-27T19:51:44Z","subjects":["Mathematics","Diamond Lemma","Free Algebras","Matrix Recognition","Matrix Relations","Noncommutative Algebra"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["hdl:1920/11246"],"render_values":[{"text":"hdl:1920/11246","href":null,"code":true}]}]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2017"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Diamond Lemma","Free Algebras","Matrix Recognition","Matrix Relations","Noncommutative Algebra"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["hdl:1920/11246"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.other","label":"Dc Description Other","values":["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."]},{"key":"dc:title","label":"Title","values":["Matrix Algebras: Equivalent Ring Relations and Special Presentations"]}]}],"canonical_facts":{"dc:date.issued":["2017"],"dc:description.other":["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:identifier":["hdl:1920/11246"],"dc:subject":["Mathematics","Diamond Lemma","Free Algebras","Matrix Recognition","Matrix Relations","Noncommutative Algebra"],"dc:title":["Matrix Algebras: Equivalent Ring Relations and Special Presentations"],"dc:type":["Dissertation"]},"updated_at":"2026-07-27T19:51:44Z"}