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 $$\mathrm {ker} \left({\phi}\right) = \left\{{0_{R_1}}\right\}$$.

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