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Is it possible to describe all possible ways in which one can define additions in the set of integers to give it a structure of ring when the multiplication is same as the usual multiplication ?

If the addition is same as the usual addition then one can easily describe all possible multiplications. What can we say when multiplication is usual ?

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    When you don't fix any structure, you just have a countable set. There is not much to say.2012-07-02
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    @Martin Brandenburg Yeah, I understand that there is not much to say about the general case. Can we say something when multiplication is usual ?2012-07-02

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