Year of Award


Document Type


Degree Type

Doctor of Philosophy (PhD)

Degree Name


Department or School/College

Department of Mathematical Sciences

Committee Chair

Adam Nyman

Commitee Members

Andrew Ware, George McRae, Jennifer Halfpap, Nikolaus Vonessen


Simple, Rank, Two-sided


University of Montana


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.



© Copyright 2009 John Walker Hart