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Uniqueness of the Coefficient Ring in Some Group Rings

Published online by Cambridge University Press:  20 November 2018

M. Parmenter  and
S. Sehgal
M. Parmenter
Affiliation:
Memorial University of Newfoundland, St. John's Newfoundland
S. Sehgal
Affiliation:
Memorial University of Newfoundland, St. John's Newfoundland
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Let 〈x〉 be an infinite cyclic group and Ri〈x〉 its group ring over a ring (with identity) Ri, for i = l and 2. Let J(Ri) be the Jacobson radical of Ri. In this note we study the question of whether or not R1〈x〉≃R2〈x〉 implies R1≃R2. We prove that this is so if Zi the centre of Ri is semi-perfect and J(Zi〈x〉) = J(Zi〈)x〉 for i = l and 2. In particular, when Zi is perfect the second condition is satisfied and the isomorphism of group rings Ri〈x〉 implies the isomorphism of Ri.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 4 , December 1973 , pp. 551 - 555
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Amitsur, S. A., Radicals of polynomial rings, Canad. J. Math. 8 (1956), 355–361.Google Scholar
2. Coleman, D. B. and Enochs, E. E., Isomorphic polynomial rings, Proc. Amer. Math. Soc. 27 (1971), 247–252.Google Scholar
3. Gilmer, R., R-automorphisms of R[x], Proc. London Math. Soc. (3) 18 (1968), 328–336.Google Scholar
4. Sehgal, S. K., Units in commutative integral group rings, Math. J. Okayama Univ. 14 (1970), 135–138.Google Scholar