Geodesic
Isomorphic Group Rings
Canadian mathematical bulletin, Tome 18 (1975) no. 4, pp. 567-576

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Let R and S be rings with 1, G a group and RG and SG the corresponding group rings. In this paper, we study the problem of when RG≃SG implies R≃S. This problem was previously investigated in [8] for the case where G is assumed to be infinite cyclic. The corresponding question for polynomial rings, namely, when does R[x]≃S[x] imply R≃S, has been considered by several authors, particularly Coleman and Enochs [3]. 
Parmenter, M. M. Isomorphic Group Rings. Canadian mathematical bulletin, Tome 18 (1975) no. 4, pp. 567-576. doi: 10.4153/CMB-1975-101-3
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