Ring Epimorphism with Trivial Kernel is Isomorphism

Theorem
Let $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$ be a ring epimorphism.

Then $\phi$ is a ring isomorphism iff $\ker \left({\phi}\right) = \left\{ {0_{R_1} }\right\}$.

Proof
Follows directly from Kernel of Monomorphism is Trivial and the definition of a ring epimorphism.