Date of Award:

8-2020

Document Type:

Thesis

Degree Name:

Master of Science (MS)

Department:

Mathematics and Statistics

Committee Chair(s)

Dariusz M. Wilczyński

Committee

Dariusz M. Wilczyński

Committee

Nathan Geer

Committee

Zhaohu Nie

Abstract

A common theme throughout algebra is the extension of arithmetic systems to ones over which new equations can be solved. For instance, someone who knows only positive numbers might think that there is no solution to x + 3 = 0, yet later learns x = -3 to be a feasible solution. Likewise, when faced with the equation 2x = 3, someone familiar only with integers may declare that there is no solution, but may later learn that x = 3/2 is a reasonable answer. Many eventually learn that the extension of real numbers to complex numbers unlocks solutions to previously unsolvable equations, such as x2 = -1.

In algebra, a ring is, roughly speaking, any arithmetic system in which addition and multiplication behave "reasonably", while a homomorphism is a function that is compatible with the appropriate arithmetic systems. Some rings are noncommutative, meaning that the order in which one multiplies may change the product (i.e. ab‚≠ ba), in contrast to most grade school arithmetic.

The extension of integers to rational numbers that allows one to solve 2x = 3 is an example of a more general technique, called localization. For commutative rings, localization is well understood and allows one to reasonably form fraction-like objects with numerators and denominators so that one can solve any equation of the form ax = b. However, this process becomes much more difficult for noncommutative rings. A modern perspective on this problem asks more broadly for an extension of a noncommutative ring which makes any given homomorphism invertible, making it possible to solve certain equations involving the homomorphism. In general, satisfactory descriptions for extensions of this type are elusive. However, there are circumstances in which it is possible to give a concrete answer.

We investigate a class of rings called the generalized triangular matrix rings whose elements are matrix-like. Our study focuses on homomorphisms whose inputs and outputs are each columns from these matrices. The results explicitly describe all of the extensions that result in available inverse homomorphisms. These extensions, called universal localizations of the ring, are also rings whose elements are matrix-like, and these matrices are more symmetric than the ring before localization. To provide some historical context, we also recount the developments in the theory that led to this research. This includes detailed descriptions of classical localization and its counterpart in noncommutative algebra, Ore localization, as well as accounts of modern viewpoints, namely Cohn localization and universal adjunction.

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