#### Year of Award

2009

#### Document Type

Dissertation

#### Degree Type

Doctor of Philosophy (PhD)

#### Degree Name

Mathematics

#### Department or School/College

Department of Mathematical Sciences

#### Committee Chair

Adam Nyman

#### Commitee Members

Andrew Ware, George McRae, Jennifer Halfpap, Nikolaus Vonessen

#### Keywords

Simple, Rank, Two-sided

#### Publisher

University of Montana

#### 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 (V^{i*},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.

#### Recommended Citation

Hart, John Walker, "SIMPLE TWO-SIDED RATIONAL VECTOR SPACES OF RANK TWO" (2009). *Graduate Student Theses, Dissertations, & Professional Papers*. 885.

http://scholarworks.umt.edu/etd/885

© Copyright 2009 John Walker Hart