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The usual addition is not the only one that makes $\mathbb{Z}$ into a ring with the usual multiplication.

Indeed, for any positive integer $n$, $\mathbb{Z}[x_1,x_2,...,x_n]$ is a unique factorization domain (UFD) with countably many primes and only $\pm{1}$ as units, so its multiplicative monoid is isomorphic to that of $\mathbb{Z}$, and the ring structure then transports along this isomorphism to one on $\mathbb{Z}$ with the usual multiplication but a different addition.

Let $A$ be the set of all binary operations $\oplus$ on $\mathbb{Z}$ for which $\oplus$ and the usual multiplication make $\mathbb{Z}$ into a ring. Then, the above shows that $A$ is infinite. But is $A$ countable or uncountable?

Note that any ring structure on $\mathbb{Z}$ with the usual multiplication must define a UFD.

Xander Henderson
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Geoffrey Trang
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    I don't think you need to look as far as $\mathbb Z[x_1, \dots, x_n]$: there are uncountably many automorphisms of $\mathbb Z$ as a multiplicative monoid, given by permuting the primes, and you can transport structure along these. – Ravi Fernando Aug 04 '22 at 16:40
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    There are uncountably many bijections from the set of primes to itself. Each gives you a different addition by transport of structure. – Arturo Magidin Aug 04 '22 at 16:40
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    Maybe you meant to ask for the number of ring stuctures up to ring isomorphism? – Jeremy Rickard Aug 04 '22 at 16:43
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    The above two arguments indicate that there are $2^{\aleph_0}$ addition operators on $\mathbb{Z}$ compatible with the multiplication monoid. – Mark Saving Aug 04 '22 at 17:05
  • I have changed the question to ask for the number of non-isomorphic ring structures instead of the number of distinct ring structures. – Geoffrey Trang Aug 21 '22 at 04:46
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    Please don’t change the question after people have addressed the original question in good faith. Why not revert this to the original form and just ask a new question? – Jeremy Rickard Aug 21 '22 at 19:41
  • Do not change your question after it has been answered. This creates a moving target for answerers. If you realize that you have left something out after you get an answer, post a new question. – Xander Henderson Aug 22 '22 at 03:31

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