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
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.
Recommended Citation
Hart, John Walker, "SIMPLE TWO-SIDED RATIONAL VECTOR SPACES OF RANK TWO" (2009). Graduate Student Theses, Dissertations, & Professional Papers. 885.
https://scholarworks.umt.edu/etd/885
© Copyright 2009 John Walker Hart