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<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>
Date25 March 2010
CreatorsHart, John Walker
ContributorsAdam Nyman, Nikolaus Vonessen, Jennifer Halfpap, George McRae, Andrew Ware
PublisherThe University of Montana
Source SetsUniversity of Montana Missoula
Detected LanguageEnglish
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