Elasticity of \(A+XI[X]\) domains where \(A\) is a Dedekind domain
type: Item
instance of: scholarly article
title: Elasticity of \(A+XI[X]\) domains where \(A\) is a Dedekind domain
zbMATH Open document ID: 1016.13001
DOI: 10.1016/S0022-4049(02)00041-5
author: Sébastien Pellerin
author: Richard Robert
published in: Journal of Pure and Applied Algebra
publication date: 2003-03-09T00:00:00Z
review text: Let \(R\) be an integral domain in which every non-zero non-unit is a product of irreducible elements. Then the elasticity of \(R\) is \[ \rho (R)=\sup \{n/m\mid x_1 \dots x_n=y_1\dots y_m;\;x_i,\;y_j\in R\text{ are irreducible}\}. \] The paper under review investigates the elasticity of integral domains of the form \(R=A+XI[X]\) where \(A\) is a Dedekind domain and \(I\) is an ideal of \(A\) that is a product of \(k\) distinct prime ideals of \(A\). It is shown that \(\rho (R)\) is finite if and only if the class group \(G\) of \(A\) is finite and in this case upper and lower bounds are given for \(\rho(R)\) involving \(k\) and certain Davenport-like constants for \(G\). In the case where \(G\) is finite and \(I\) is a product of \(k\) distinct principal primes, the inequalities reduce to \(\rho(R) =k+(D(G)- 1)/2\) where \(D(G)\) is the Davenport constant of \(G\). This generalizes a similar result of \textit{N. Gonzales}, \textit{S. Pellerin} and \textit{R. Robert} [J. Pure Appl. Algebra 160, 183-194 (2001; Zbl 1001.13008)] in the case where \(A\) is a PID.
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13B24
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13C20
zbMATH DE Number: 1878440
zbMATH Keywords: going up ring
zbMATH Keywords: class group
zbMATH Keywords: elasticity of integral domain
zbMATH Keywords: Dedekind domain
reviewed by: D. D. Anderson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1878440