The purpose of this thesis is to find sufficient conditions under which a non-commutative version of the polynomial ring in two variables exists. The non-commutative rings we construct are non-commutative symmetric algebras over a two-sided vector space. After reviewing the definition of a two-sided vector space and giving some examples, we briefly recall the theory of simple two-sided vector spaces. We then assume k is a field of characteristic zero and t is transcendental over k and we find sufficient conditions under which a simple k-central two-sided vector space V over k(t) has left and right dimension two. Given such a V, and letting *V and V* denote the left and right duals we find conditions under which (Vi*,V(i+1)*,V(i+2)* ) has a simultaneous for all i, i an integer. This condition implies the non-commutative symmetric algebra over V can be constructed. We conclude by exhibiting a five-dimensional family of simple k-central two-sided vector spaces over k(t) of left and right dimension two who non-commutative symmetric algebras exist.
","abstract_html":"<p>The purpose of this thesis is to find sufficient conditions under which a non-commutative version of the polynomial ring in two variables exists. The non-commutative rings we construct are non-commutative symmetric algebras over a two-sided vector space. After reviewing the definition of a two-sided vector space and giving some examples, we briefly recall the theory of simple two-sided vector spaces. We then assume k is a field of characteristic zero and t is transcendental over k and we find sufficient conditions under which a simple k-central two-sided vector space V over k(t) has left and right dimension two. Given such a V, and letting <sup>*</sup>V and V<sup>*</sup> denote the left and right duals we find conditions under which (V<sup>i*</sup>,V<sup>(i+1)*</sup>,V<sup>(i+2)*</sup> ) has a simultaneous for all i, i an integer. This condition implies the non-commutative symmetric algebra over V can be constructed. We conclude by exhibiting a five-dimensional family of simple k-central two-sided vector spaces over k(t) of left and right dimension two who non-commutative symmetric algebras exist.</p>","abstract_has_math":false,"creators":["Hart, John Walker"],"institution":"University of Montana","degree_name":"Doctor of Philosophy (PhD)","degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-01-01T08:00:00Z","date_published":"2009-01-01T08:00:00Z","updated_at":"2026-07-24T03:13:58Z","subjects":["Simple","Rank","Two-sided"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.umt.edu/etd/885","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Hart, John Walker"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["University of Montana"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Simple","Rank","Two-sided"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.umt.edu/etd/885"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The purpose of this thesis is to find sufficient conditions under which a non-commutative version of the polynomial ring in two variables exists. The non-commutative rings we construct are non-commutative symmetric algebras over a two-sided vector space. After reviewing the definition of a two-sided vector space and giving some examples, we briefly recall the theory of simple two-sided vector spaces. We then assume k is a field of characteristic zero and t is transcendental over k and we find sufficient conditions under which a simple k-central two-sided vector space V over k(t) has left and right dimension two. Given such a V, and letting *V and V* denote the left and right duals we find conditions under which (Vi*,V(i+1)*,V(i+2)* ) has a simultaneous for all i, i an integer. This condition implies the non-commutative symmetric algebra over V can be constructed. We conclude by exhibiting a five-dimensional family of simple k-central two-sided vector spaces over k(t) of left and right dimension two who non-commutative symmetric algebras exist.
"]},{"key":"dc:title","label":"Title","values":["SIMPLE TWO-SIDED RATIONAL VECTOR SPACES OF RANK TWO"]}]}],"canonical_facts":{"dc:creator":["Hart, John Walker"],"dc:description.abstract":["The purpose of this thesis is to find sufficient conditions under which a non-commutative version of the polynomial ring in two variables exists. The non-commutative rings we construct are non-commutative symmetric algebras over a two-sided vector space. After reviewing the definition of a two-sided vector space and giving some examples, we briefly recall the theory of simple two-sided vector spaces. We then assume k is a field of characteristic zero and t is transcendental over k and we find sufficient conditions under which a simple k-central two-sided vector space V over k(t) has left and right dimension two. Given such a V, and letting *V and V* denote the left and right duals we find conditions under which (Vi*,V(i+1)*,V(i+2)* ) has a simultaneous for all i, i an integer. This condition implies the non-commutative symmetric algebra over V can be constructed. We conclude by exhibiting a five-dimensional family of simple k-central two-sided vector spaces over k(t) of left and right dimension two who non-commutative symmetric algebras exist.
"],"dc:identifier":["https://scholarworks.umt.edu/etd/885"],"dc:publisher":["University of Montana"],"dc:subject":["Simple","Rank","Two-sided"],"dc:title":["SIMPLE TWO-SIDED RATIONAL VECTOR SPACES OF RANK TWO"],"dc:type":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:13:58Z"}