Characterization of Group of Invertible Elements of Six Index Zero Completely Primary Finite Rings of Characteristic p

Abstract

The study of finite extension of Galois rings in the recent past have given rise to commutative completely primary finite rings that have attracted much attention as they have yielded important results towards classification of finite rings into well-known structures. In this paper, we give a construction of a class of completely primary finite ring R of characteristic p whose subsets of zero divisors ( ) Z R satisfy the condition ( ) ( ) ( ) ( ) ( ) ( ) 6 5 0 ; 0 Z R Z R =  . The ring R is constructed over its subring ( ) 0 ,r R GR p p = as an idealization of the 0 R- modules. A thorough determination and classification of the structure of the group of invertible elements using fundamental theorem of finitely generated abelian groups is given.