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Isomorphic Group Rings with Non-Isomorphic Coefficient Rings*

Published online by Cambridge University Press:  20 November 2018

L. Grünenfelder  and
M. M. Parmenter
L. Grünenfelder
Affiliation:
Dalhousie University, HalifaxNova ScotiaCanadaB3H 4H8
M. M. Parmenter
Affiliation:
Memorial University of Newfoundland, St. John'sNewfoundlandCanadaA1B 3X7
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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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 2 , 01 June 1980 , pp. 245 - 246
Copyright
Copyright © Canadian Mathematical Society 1980

Footnotes

*

Research supported in part by the National Research Council of Canada, grants no. A-8741 and A-8775.

References

1. Mislin, G., Nilpotent groups with finite commutator subgroups.Springer Lecture Notes 418 (1974).Google Scholar
2. Sandling, R., Group rings of circle and unit groups. Math. Z. 140 (1974), 195-202.Google Scholar
3. Sehgal, S. K., On the isomorphism of integral group rings II. Can. J. Math. 21 (1969), 1182-1188.Google Scholar
4. Sehgal, S. K., Topics in group rings. Marcel Dekker (1978).Google Scholar
5. Walker, E. A., Cancellation in direct sums of groups. Proc. Am. Math. Soc. 7 (1956), 898-902Google Scholar