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I have a set $R$ with binary operations $\oplus$ and $\otimes$. I'm pretty certain my set $R$ together with these two binary operations forms a ring, and I have already proved that $\phi:R\to\mathbb{Z}$ is a ring homomorphism, where $\phi$ is some function.

The thing is I haven't actually shown that $R$ is a ring yet (although I'm pretty sure it is)

Do I need to show that $R$ is a ring, or does the existence of the ring homomorphism together with knowledge that $\mathbb{Z}$ is a ring imply that $R$ is a ring somehow?

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Ring homomorphisms are only defined between things that we already know are rings. What you have is a function that respects two binary operations, which is not sufficient. Here is a counterexample. Consider the set $\mathbb{N}$ with addition and multiplication. This is not a ring because there are no additive inverses. But the natural inclusion $\mathbb{N}\to\mathbb{Z}$ respects addition and multiplication.

Now, if you could find a function that preserved your operations and were a bijection of sets, that would be sufficient (think about why).

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