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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 R1 and R2 be commutative rings with identities, G a group and R1G and R2G the group ring of G over R1 and R2 respectively. The problem that motivates this work is to determine what relations exist between R1 and R2 if R1G and R2G are isomorphic. For example, is the coefficient ring R1 an invariant of R1G? This is not true in general as the following example shows. Let H be a group and

If R1 is a commutative ring with identity and R2 = R1H, then

but R1 needn't be isomorphic to R2.

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.

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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