Elasticity of \(A+XI[X]\) domains where \(A\) is a Dedekind domain
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.
scientific article; zbMATH DE number 1878440