Kernel is Trivial iff Monomorphism

Theorem

Kernel of Group Monomorphism

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a group homomorphism.

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

Then $\phi$ is a group monomorphism if and only if $\map \ker \phi$ is trivial.

Kernel of Ring Monomorphism

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 if and only if $\map \ker \phi = 0_{R_1}$.