Kernel of Monomorphism is Trivial

Theorem

Kernel of Group Monomorphism

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a group homomorphism.

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

Then $\phi$ is a group monomorphism iff $\ker \left({\phi}\right)$ is trivial.

Kernel of Ring Monomorphism

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