Inclusion Mapping on Subring is Homomorphism

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 homomorphism.

Proof
Let $x, y \in S$.

Then:

and:

Hence $\iota$ is a ring homomorphism by definition.