# Can I prove a set with operations is a ring via a ring homomorphism?

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?

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.