Canonical Homomorphism to Polynomial Ring is Ring Monomorphism
Contents
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
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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 $h : R[X] \to R$ 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.
$\blacksquare$
