On the Structure of a Class of Galois Ring Module Idealization

Abstract

Let R_o be a Galois ring and U is a finitely generated R_o- module. Consider an idealization of U expressed as R = R_o \(\bigoplus\) U endowed with a suitable multiplication. We explore the structure of R through its group of units R^x and the graph of its zero divisors \(\Gamma\)(R). The study involves an investigation on the overarching interplay between the ring theoretical properties of R, the group theoretic properties of R^x and the graph theoretic properties of \(\Gamma\)(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.