Canonical Homomorphism to Polynomial Ring is Ring Monomorphism

Theorem

Let $R$ be a commutative ring with unity.

Let $\struct {R \sqbrk X, \iota, X}$ be a polynomial ring over $R$ in one indeterminate $X$.

Then the canonical homomorphism $\iota : R \to R \sqbrk X$ is a ring monomorphism.

Proof using explicit construction

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Proof using universal property

Outline of Proof

We apply the Universal Property of Polynomial Ring to construct a left inverse of $\iota$.

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 $h : R \sqbrk X \to R$ with $h \circ \iota = \operatorname{id}$.

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

By Injection if Composite is Injection, $\iota$ is an injection.

Thus $\iota$ is a ring monomorphism.

$\blacksquare$