Geodesic
Isomorphic Group Rings with Non-Isomorphic Coefficient Rings*
Canadian mathematical bulletin, Tome 23 (1980) no. 2, pp. 245-246

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The following question has been floating around for some time now and is also stated as Research Problem 26 in [4]:Let R, S be unital rings and let 〈x〉 be an infinite cyclic group. Does R〈x〉≃S〈x〉 imply R≃S?In this note, we present a collection of examples which answer the question in the negative. However, all of these examples consist of non-commutative rings, and the problem is still open in the case where R and S are assumed to be commutative. 
Grünenfelder, L.; Parmenter, M. M. Isomorphic Group Rings with Non-Isomorphic Coefficient Rings*. Canadian mathematical bulletin, Tome 23 (1980) no. 2, pp. 245-246. doi: 10.4153/CMB-1980-034-4
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     author = {Gr\"unenfelder, L. and Parmenter, M. M.},
     title = {Isomorphic {Group} {Rings} with {Non-Isomorphic} {Coefficient} {Rings*}},
     journal = {Canadian mathematical bulletin},
     pages = {245--246},
     year = {1980},
     volume = {23},
     number = {2},
     doi = {10.4153/CMB-1980-034-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1980-034-4/}
}
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