Suppose that $ϕ : R \to S$ is a ring isomorphism. As $ϕ$ is a bijection, it has an inverse map $ψ : S \to R$ such that $ψ \circ ϕ : R \to R$ is the identity map on $R$ and $ϕ \circ ψ : S \to S$ is the identity map on $S$. Prove that $ψ : S \to R$ is a ring homomorphism.
I've proved this one way, but I need to prove that it is an isomorphism given that it's a ring homomorphism.
Any help would be very much appreciated
The ring homomorphism $\psi$ is a ring isomorphism since it is a bijection. You know that because you know that $\psi^{-1}=\phi$.