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

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DOI

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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     author = {Parmenter, M. M.},
     title = {Isomorphic {Group} {Rings}},
     journal = {Canadian mathematical bulletin},
     pages = {567--576},
     year = {1975},
     volume = {18},
     number = {4},
     doi = {10.4153/CMB-1975-101-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1975-101-3/}
}
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