Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-09-01T02:04:45.001Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let 〈x〉 be an infinite cyclic group and R i 〈x〉 its group ring over a ring (with identity) R i , for i = l and 2. Let J(R i ) be the Jacobson radical of R i . In this note we study the question of whether or not R 1〈x〉≃R 2〈x〉 implies R 1≃R 2. We prove that this is so if Z i the centre of R i is semi-perfect and J(Z i 〈x〉) = J(Z i 〈)x〉 for i = l and 2. In particular, when Z i is perfect the second condition is satisfied and the isomorphism of group rings R i 〈x〉 implies the isomorphism of R i .

Information

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