Kernel is Trivial iff Monomorphism/Ring

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

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

Then $\phi$ is a ring monomorphism $\map \ker \phi = 0_{R_1}$.

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

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