Date of Award

Spring 2024

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Edward Hinson

Second Advisor

Maria Basterra

Third Advisor

Cain Edie-Michell

Abstract

For a commutative ring $R$ with identity, the set of unimodular vectors $Um_n(R)$ contains all column vectors $\alpha\in R^n$ so that $\alpha^T\beta=1$ for some $\beta\in R^n$. The set of completable vectors $Umc_n(R)$ is the set of unimodular vectors in $R^n$ that appear as the first column of an invertible matrix in $GL_n(R)$. Determining the conditions in which unimodular vectors are completable is fundamental to problems in algebraic geometry and module theory. In this work, we generalize the usual dimension $3$, real vector cross product to commutative rings and characterize all completable vectors in $Umc_3(R)$ in terms of the cross product. We exhibit an application of the cross product by using it to represent invertible matrices and traverse path components of $Um_3(R)$. An action by the units of $R$ on $Um_n(R)/E_n(R)$ is then introduced and we show that this action acts trivially on elementary orbits whose elements have unimodular kernel elements. This provides a means of determining that a unimodular vector is not completable by showing this action is nontrivial on its elementary orbit. We then connect the study of this action to the study of eigenvectors of elementary matrices.

Download

Share

COinS