## Theses and Dissertations

#### Title

On invertible algebras

#### DOI

10.17077/etd.ts52-wz3k

Dissertation

Spring 2019

#### Access Restrictions

Access restricted until 07/29/2020

#### Degree Name

PhD (Doctor of Philosophy)

Mathematics

#### First Advisor

Iovanov, Miodrag C

Bleher, Frauke

Camillo, Victor

Kinser, Ryan

Muhly, Paul

#### Abstract

An algebra $A$ over a field $\K$ is said to be \textit{invertible} if it has a basis $\B$ consisting only of of units; if $\B^{-1}$ is again a basis, $A$ is \textit{invertible-2}, or \textit{I2}. The question of when an invertible algebra is necessarily I2 arises naturally. The study of these algebras was recently initiated by Lòpez-Permouth, Moore, Szabo, Pilewski \cite{lopezIJM}, \cite{lopez1}. In this work, we prove several positive results on this problem, answering also some questions and generalizing a few results from these papers. We show that every field is an I2 algebra over any subfield, and that any subalgebra of the rational functions field $K(x)$ that strictly contains $K[x]$, with $K$ an algebraically closed field, has a symmetric basis $\B=\B^{-1}$. Using this, we expand the class of examples of algebras known to be invertible or I2 with several classes, such as semiprimary rings over fields $K\neq \F_2$ satisfying some additional mild conditions. We also show that every commutative, finitely generated, invertible algebra without zero divisors is almost I2 in the sense that it becomes I2 after localization at a single element.

#### Keywords

good ring, I2 algebra, invertible algebra, invertible ring, sum of units

vii, 72 pages

#### Bibliography

Includes bibliographical references (pages 71-72).

#### Copyright

Copyright © 2019 Jeremy R. Edison

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