Inclusion Mapping on Subring is Monomorphism

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 monomorphism.

Proof

By Inclusion Mapping on Subring is Homomorphism, $i_S$ is a ring homomorphism.

By Inclusion Mapping is Injection, $i_S$ is an injection.

The result follows by definition of (ring) monomorphism.

$\blacksquare$