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If we define a ring on the integers $\mathbb Z$ (i.e. $(\mathbb Z, +,\times)$) and equip it with the usual $\times$ operation for $\mathbb Z$, is it necessary that the $+$ operation be the usual $+$ too?

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up vote 14 down vote accepted

The monoid $(\mathbb Z\setminus\{0\},\times)$ is isomorphic to the direct sum $\mathbb Z_2\oplus\bigoplus\limits_{\text{countable}}\mathbb N_0$ of a cyclic group of order $2$ and a countable number of copies of $\mathbb N_0$. It follows that it has lots of automorphisms: we can permute the prime numbers arbitrarily.

If $\phi:\mathbb Z\to\mathbb Z$ is any such automorphism which is not the identity, then we get a ring structure on it by defining a new sum $a+'b=\phi^{-1}(\phi(a)+\phi(b))$. The uncountably many rings we get this way are of course isomorphic to the usual $\mathbb Z$, but all different.

Later. A different direction one can go is the following:

The polynomial ring $\mathbb F_3[X]$ is a principal ideal domain with countably infinite many primes and exactly two invertible elements, so its monoid of non-zero elements is isomorphic to $(\mathbb Z\setminus\{0\},\times)$. There is then a bijection $f:\mathbb F_3[X]\to\mathbb Z$ mapping zero to zero which is an isomorphism of monoids in the complements of those zero elements. We can define an addition on $\mathbb Z$ by transporting that of $\mathbb F_3[X]$ along $f$. This turns $\mathbb Z$ into a ring of characteristic $3$ :)

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Thanks Mariano! – Jason H Mar 4 '12 at 9:12

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