On quotients of generalized Euclidean group rings

ABSTRACT

Let R = ℤ[C] be the integral group ring of a finite cyclic group C. Dennis et al. [4] proved that R is a generalized Euclidean ring in the sense of Cohn [3], 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.

KEYWORDS:

• Elementary matrix
• generalized Euclidean ring
• group ring
• metabelian group
• Nielsen equivalence
• quotient
• unimodular row

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 13F07
• Secondary 16S34

Acknowledgment

The author thanks Pierre de la Harpe and Tatiana Smirnova-Nagnibeda for encouragements and comments made on preliminary versions of this paper.