The Zero Divisor Graphs of Specific Commutative Rings

Abstract

Let A be a commutative ring with 1. In 1998, David F. Anderson and Philip S. Livingston associated to A a graph Γ(A) and they called it the zero divisor graph of A. The vertices of Γ(A) is the set 〖Z(A)〗^*= Z(A) - {0}, where Z(A) denotes the set of all zero divisors of A, and for x ≠ y in 〖Z(A)〗^*, the vertices x and y are adjacent if and only if xy = 0 [3]. In this thesis, we provide a study of the effect of some basic ring theoretic properties of a ring A on it’s zero divisor graph (Γ(A)) by reproducing and illustrating using new examples, the main work done in [3, 12]. Moreover, in the last chapter, we investigate for the first time, the interplay between the ring-theoretic properties of some special rings; such as Boolean, K-Boolean, and nilpotent rings; and the graph theoretic properties of their zero divisor graphs.