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Is there any requirement that the two operations of a ring have to be related to each other, excluding the requirement of distributivity? We all know from grade school that multiplication of integers is repeated addition, and we also know that the integers form a ring under addition and multiplication. Are there any other examples of rings whose second operation is based off of and/or defined in terms of the first, or are the integers basically the only example of that?

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    $\begingroup$ The integers modulo $m$ come close. $\endgroup$ – André Nicolas Feb 4 '16 at 19:09
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Is there any requirement that the two operations of a ring have to be related to each other, excluding the requirement of distributivity?

The only requirements on what a ring is are given in the axioms for a ring, and if something does not appear there, there is no requirement for it.

Are there any other examples of rings whose second operation is based off of and/or defined in terms of the first, or are the integers basically the only example of that?

I think the integers are unique with respect to this property. Its quotients too, probably, if you are willing to interpret repeated additions as multiplication by cosets.

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