Kernel is Trivial iff Monomorphism/Ring

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

Let $\ker \left({\phi}\right)$ be the kernel of $\phi$.

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

Proof
The proof for the ring monomorphism follows directly from:
 * Ring Homomorphism of Addition is Group Homomorphism

and:
 * Kernel of Group Monomorphism for the group monomorphism.