Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism

Definition
Let $\left({R, +, \circ}\right)$ and $\left({S, \oplus, *}\right)$ be rings.

Let $\phi: R \to S$ be a (ring) homomorphism.

Then $\phi$ is a ring isomorphism iff $\phi$ is a bijection. That is, $\phi$ is a ring isomorphism iff $\phi$ is both a monomorphism and an epimorphism.