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Abstract

In this paper, we examine the algebraic properties of localizations of commutative rings and how localizations affect the zero-divisor graphs structure of modular rings. We also classify the zero-divisor graphs of modular rings with respect to both the diameter and girth of their resultant zero-divisor graphs.

Author Bio

Tom Cuchta is a senior mathematics and applied mathematics major at Marshall University who should be graduating in Spring 2009. He plans on pursuing a PhD after graduating. This paper is part of the summer research that he completed at Wabash College over the summer of 2008 -- it will be presented at the National AMS/MAA Meetings in January 2009. When not doing mathematics, Tom enjoys playing the bassoon.

Kathryn A. Lokken is a mathematics major at the University of Wisconsin-Madison. She will be graduating in Spring 2009 and intends to earn a PhD after graduation. This paper is part of the research that she completed at Wabash College over the summer of 2008 and it will be presented, in poster form, at the National AMS/MAA Meetings in January 2009.

William Young is a first-year graduate student at Vanderbilt Univesity. His research for this paper was done at Wabash College for an REU while an undergraduate at Purdue University in the summer of 2007. He presented portions of it at the joint meeting of the AMS and MAA in January 2008. William's non-mathematical interests include the Supreme Court, philosophy, and anthropology.

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