On the Structure of a Class of Galois Ring Module Idealization

Let Ro be a Galois ring and U is a finitely generated Ro- module. Consider an idealization of U expressed as R = Ro
U endowed with a suitable multiplication. We explore the structure of R through its group of units Rx and the graph of its zero divisors
(R). The study involves an investigation on the overarching interplay between the ring theoretical properties of R, the group theoretic properties of Rx and the graph theoretic properties of
(R). Since R is a finite ring with identity, the convention that each element of R is either a unit or a zero divisor has been extensively used to drive the concept of classification of the elements of R. The units of R have been classified, the automorphisms of R have been determined and the zero divisors of R have been characterized.