\begin{definition}[Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism]
Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ 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.
\end{definition}