Ring Epimorphism with Trivial Kernel is Isomorphism

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

Let $K = \map \ker \phi$ be the kernel of $\phi$.

$\phi$ is an isomorphism $K = \set {0_{R_1} }$.

Proof
From Kernel is Trivial iff Monomorphism, $\phi$ is a ring monomorphism $K = \set {0_{R_1} }$.

As $\phi$ is also an epimorphism, the result follows.