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On the Uniqueness of the Coefficient Ring in a Group Ring

Published online by Cambridge University Press:  20 November 2018

Isabelle Adjaero  and
Eugene Spiegel
Isabelle Adjaero
Affiliation:
University of Nigeria – Nsukka, Ahambra State, Nigeria
Eugene Spiegel
Affiliation:
University of Connecticut, Storrs, Connecticut
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Let R 1 and R 2 be commutative rings with identities, G a group and R 1 G and R 2 G the group ring of G over R 1 and R 2 respectively. The problem that motivates this work is to determine what relations exist between R 1 and R 2 if R 1 G and R 2 G are isomorphic. For example, is the coefficient ring R 1 an invariant of R 1 G? This is not true in general as the following example shows. Let H be a group and

If R 1 is a commutative ring with identity and R 2 = R 1 H, then

but R 1 needn't be isomorphic to R 2.

Several authors have investigated the problem when G = <x>, the infinite cyclic group, partly because of its closeness to R[x], the ring of polynomials over R.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 4 , 01 August 1983 , pp. 654 - 673
Copyright
Copyright © Canadian Mathematical Society 1983

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