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Isomorphic Group Rings of Free Abelian Groups

Published online by Cambridge University Press:  20 November 2018

Jan Krempa
Jan Krempa*
Affiliation:
University of Warsaw, Warsaw, Poland
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S. K. Sehgal ([9], Problem 26) proposed the following question : Let A, B be rings and X an infinite cyclic group. Does AX ⋍ BX imply A ⋍ B? M. M. Parmenter and S. K. Sehgal (cf. [9], Chapter 4) proved that, under some strong assumptions concerning rings A, B, the answer is affirmative. In this paper, we show that the assumptions concerning the ring B may be omitted in the above mentioned results. Moreover, it is proven that if (AX)X ⋍ BX then AX ⋍ B for all rings A, B. If A is commutative and noetherian then a partial answer to Problem 27, [9] follows from our results.

Recently, L. Griinenfelder and M. M. Parmenter constructed nonisomorphic rings A, B for which the group rings AX, BX are isomorphic, [2], We give a new class of rings of this type. Our examples are noncommutative domains and algebras over finite fields. That also gives a negative answer to Problem 29, [9].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 1 , 01 February 1982 , pp. 8 - 16
Copyright
Copyright © Canadian Mathematical Society 1982

References

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