Congruences in \({\mathbb{Z}}^ n\), finite Abelian groups and the Chinese remainder theorem
type: Item
instance of: scholarly article
title: Congruences in \({\mathbb{Z}}^ n\), finite Abelian groups and the Chinese remainder theorem
zbMATH Open document ID: 0631.10007
DOI: 10.1016/0012-365X(87)90170-1
author: B. George
published in: Discrete Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let \({\mathbb{Z}}^ n\) denote the additive group of column n-vectors with integral coordinates and \({\mathbb{Z}}^{n\times n}\) the ring of \(n\times n\) integral matrices. If \(M\in {\mathbb{Z}}^{n\times n}\) is a regular matrix, then the author generalizes the concept of the congruence to \({\mathbb{Z}}^ n\) in saying that \({\mathfrak a}\equiv {\mathfrak b}\) (mod M) for \({\mathfrak a}\equiv {\mathfrak b}\in {\mathbb{Z}}^ n\) if the difference \({\mathfrak a}-{\mathfrak b}\) belongs to \(M{\mathbb{Z}}^ n\), the set of all linear integral combinations of column vectors of M. He shows, using the Smith normal form theorem for M, that every finite Abelian group G is realizable as the quotient group \({\mathbb{Z}}^ n/M{\mathbb{Z}}^ n\) of integral vectors modulo M. As a by-product a simple proof of the Chinese remainder theorem is given.
Mathematics Subject Classification ID: 11A99
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11C99
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 4025504
zbMATH Keywords: congruence in \(Z^ n\)
zbMATH Keywords: group of integral vectors
zbMATH Keywords: integral matrices
zbMATH Keywords: finite Abelian group
zbMATH Keywords: Chinese remainder theorem
reviewed by: Štefan Porubský
MaRDI profile type: Publication
cites work: Q4101884
cites work: Two theorems on matrices
description: scientific article; zbMATH DE number 4025504