On quotients of generalized Euclidean group rings

TL;DR

This paper proves that all proper quotients of the integral group ring of a finite cyclic group are also generalized Euclidean rings, extending previous results about the ring itself.

Contribution

It establishes that every proper quotient of the integral group ring of a finite cyclic group retains the generalized Euclidean property.

Findings

Proper quotients of the ring are generalized Euclidean.

Extends known results from the ring to its quotients.

Supports the structure of elementary matrices in quotients.

Abstract

Let $R = Z[C]$ be the integral group ring of a finite cyclic group $C$. Dennis and al. proved that $R$ is a generalized Euclidean ring in the sense of P. M. Cohn, i.e., $SL_n(R)$ is generated by the elementary matrices for all $n$. We prove that every proper quotient of $R$ is also a generalized Euclidean ring.