Canonical Homomorphism to Polynomial Ring is Ring Monomorphism

Theorem
Let $R$ be a commutative ring with unity.

Let $(S, f, X)$ be a polynomial ring over $R$ in one indeterminate $X$.

Then the canonical homomorphism $f : R \to S$ is a ring monomorphism.

Outline of Proof
We apply the Universal Property of Polynomial Ring to construct a left inverse of $f$.

Proof
Let $\operatorname{id} : R \to R$ be the identity mapping.

Let $1$ be the unity of $R$.

By Identity Mapping is Ring Automorphism, $\operatorname{id}$ is a ring homomorphism.

By Universal Property of Polynomial Ring, there exists a ring homomorphism $h : S \to R$ with $h\circ f = \operatorname{id}$.

By Identity Mapping is Injection, $\operatorname{id}$ is an injection.

By Injection if Composite is Injection, $f$ is an injection.

Thus $f$ is a ring monomorphism.