On the Adjacency Matrices of the Anderson-Livingston Zero Divisor Graphs of Galois Rings

Abstract

A ring is Galois if its subset of all the zero divisors (including zero) forms a principal ideal. Galois rings are generalizations of Galois fields and have been used widely in the past few decades to construct various optimal families of error correcting codes. These rings are important in the structure theory of finite commutative rings. Furthermore, Ga lois rings are the building blocks of Completely Primary Finite Rings (C.P.F.R) which play a fundamental role in the study of the structures of finite rings in the sense that every finite ring is expressible as a di rect sum of matrix rings over Completely Primary Finite Rings. Zero divisor graphs have been of interest to Mathematicians, particularly in understanding the structures of zero divisors in finite rings. However, not much is known about the adjacency matrices of zero divisor graphs of finite rings. In this paper we present some explicit results on the adjacency matrices of the Anderson-Livingston zero divisor graphs of Galois rings