Zero-Divisor Graphs of Direct Product Rings

Abstract

This research investigates zero-divisor graphs of direct product rings. In zero-divisor graphs, if elements are connected with an edge, they will multiply to the additive identity element of direct product rings, (0,0). We look at graphs of the form Z_2p×Z_2q, where p and q are prime numbers greater than 2. In order to generalize the structure of these graphs, many specific examples of Z_2p×Z_2q graphs are analyzed to better understand the common form. Upon investigating these rings, many “clumps”, or “families”, of zero-divisors appeared. Certain families will always connect to other families, and some families will never connect. This research investigates which families connect, making it possible to generalize the structure of these graphs.