Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring

Sami Alabiad,Yousef Alkhamees

doi:10.3390/sym13020307

Abstract

A finite ring with an identity whose lattice of ideals forms a unique chain is called a finite chain ring. Let R be a commutative chain ring with invariants p,n,r,k,m. It is known that R is an Eisenstein extension of degree k of a Galois ring S=GR(pn,r). If p−1 does not divide k, the structure of the unit group U(R) is known. The case (p−1)∣k was partially considered by M. Luis (1991) by providing counterexamples demonstrated that the results of Ayoub failed to capture the direct decomposition of U(R). In this article, we manage to determine the structure of U(R) when (p−1)∣k by fixing Ayoub’s approach. We also sharpen our results by introducing a system of generators for the unit group and enumerating the generators of the same order.

Highlights

IntroductionWe consider finite commuative chain rings, some results still correct under more general situation
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
One major result claims that the factorization of an abelian p−group with incomplete j-diagram can be completely obtained by the mentioned diagram

Summary

Introduction

We consider finite commuative chain rings, some results still correct under more general situation. Ayoub called these rings (cf [1]) primary homogeneous rings. One major result claims that the factorization of an abelian p−group with incomplete j-diagram can be completely obtained by the mentioned diagram This idea was used to find the structure of U ( R) when R is not necessarily finite (Theorem 3, Section 4 [1]). With this relation being taken into consideration, we manage to make incomplete j-diagrams succeed in recapturing the structure of multiplcative groups of finite chain rings.

Preliminaries and Notations

The Determination of the Structure of H

Enumeration of Generators of the Same Order