Isomorphic Group Rings with Non-Isomorphic Coefficient Rings*

Abstract

The following question has been floating around for some time now and is also stated as Research Problem 26 in [4]:Let R, S be unital rings and let 〈x〉 be an infinite cyclic group. Does R〈x〉≃S〈x〉 imply R≃S?In this note, we present a collection of examples which answer the question in the negative. However, all of these examples consist of non-commutative rings, and the problem is still open in the case where R and S are assumed to be commutative.