Canonical Homomorphism to Polynomial Ring is Ring Monomorphism

From ProofWiki
Jump to: navigation, search

Theorem

Let $R$ be a commutative ring with unity.

Let $(R[X], \iota, X)$ be a polynomial ring over $R$ in one indeterminate $X$.


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


Proof using explicit construction


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 $h : R[X] \to R$ 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.

$\blacksquare$