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Aug 31, 2011 at 19:30 comment added Pace Nielsen Nice example. I suppose the 'nicest' example of a different ring structure on $\Q$ would be to take Cantor's bijection $f:\mathbb{Q}\rightarrow \mathbb{Z}$ and then apply Amit's answer, using the usual ring structure on $\mathbb{Z}$.
Aug 31, 2011 at 19:07 history answered Noah Stein CC BY-SA 3.0