Geodesic
A Problem of Herstein on Group Rings
Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 201-202

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Let F be a field of characteristic 0 and G a group such that each element of the group ring F[G] is either (right) invertible or a (left) zero divisor. Then G is locally finite.This answers a question of Herstein [1, p. 36] [2, p. 450] in the characteristic 0 case. The proof can be informally summarized as follows: Let g l,...,gn be a finite subset of G, and let 1—x is not a zero divisor so it is invertible and its inverse is 1+x+x 3+⋯. The fact that this series converges to an element of F[G] (a finite sum) forces the subgroup generated by g 1,...,gn to be finite, proving the theorem. The formal proof is via epsilontics and takes place inside of F[G]. 
Formanek, Edward. A Problem of Herstein on Group Rings. Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 201-202. doi: 10.4153/CMB-1974-040-9
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     title = {A {Problem} of {Herstein} on {Group} {Rings}},
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     year = {1974},
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     number = {2},
     doi = {10.4153/CMB-1974-040-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-040-9/}
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