Inclusion Mapping on Subring is Monomorphism

Theorem
Let $\struct {R, +, \circ}$ be an Ring.

Let $\struct {S, +_{\restriction S}, \circ_{\restriction S}}$ be a subring of $R$.

Let $\iota: S \to R$ be the inclusion mapping from $S$ to $R$.

Then $\iota$ is a ring monomorphism.

Proof
By Inclusion Mapping on Subring is Homomorphism, $\iota$ is a ring homomorphism.

By Inclusion Mapping is Injection, $\iota$ is an injection.

The result follows by definition of (ring) monomorphism.