Inclusion Mapping on Subring is Homomorphism

Theorem
Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +{\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$.

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

Then ${i_S}$ is a ring homomorphism.

Proof
Let $x, y \in S$.

Then:

and:

Hence ${i_S}$ is a ring homomorphism by definition.