On the groups of units of finite commutative chain rings.
type: Item
instance of: scholarly article
title: On the groups of units of finite commutative chain rings.
zbMATH Open document ID: 1088.13010
DOI: 10.1016/S1071-5797(02)00003-5
author: Ka Hin Leung
author: Xiang-Dong Hou
author: Siu Lun Ma
published in: Finite Fields and their Applications
publication date: 2003-04-02T00:00:00Z
review text: A commutative chain ring is a commutative ring whose ideals form a chain under set theoretic inclusion. Let \(M\) be the maximal ideal of a finite commutative chain ring \(R\) with characteristic \(p^n\), then \(R/M\) is a finite field, say \(R/M \cong \text{GF}(p^r)\), and \(pR=M^e\), \(e \leq s\), where \(s\) is the nilpotency of \(M\). When \(p-1\) is not a divisor of \(e\), the structure of the group of units of \(R\) is known (which depends only on the parameters \(p,n,r,e,s\)). In the paper, the authors give an algorithm to compute the structure of the group of units when \(p-1\) is a divisor of \(e\). Such a structure depends on the parameters \(p,n,r,e,s\), and the Eisenstein polynomial that defines \(R\) as an extension over the Galois ring \(\text{GR}(p^n,r)\). The authors also refine the known result for the case \((p-1) \nmid e\) by listing a set of linearly independent generators for the group of units of \(R\).
Mathematics Subject Classification ID: 13M05
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 11T30
zbMATH DE Number: 1891472
zbMATH Keywords: finite commutative chain ring
zbMATH Keywords: group of units
zbMATH Keywords: Galois ring
zbMATH Keywords: Eisenstein polynomial
reviewed by: Hai Quang Dinh
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1891472