Geodesic
Uniqueness of the Coefficient Ring in Some Group Rings
Canadian mathematical bulletin, Tome 16 (1973) no. 4, pp. 551-555

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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 . 
Parmenter, M.; Sehgal, S. Uniqueness of the Coefficient Ring in Some Group Rings. Canadian mathematical bulletin, Tome 16 (1973) no. 4, pp. 551-555. doi: 10.4153/CMB-1973-090-5
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     author = {Parmenter, M. and Sehgal, S.},
     title = {Uniqueness of the {Coefficient} {Ring} in {Some} {Group} {Rings}},
     journal = {Canadian mathematical bulletin},
     pages = {551--555},
     year = {1973},
     volume = {16},
     number = {4},
     doi = {10.4153/CMB-1973-090-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1973-090-5/}
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