# Uniqueness of Polynomial Ring in One Variable

## Theorem

Let $R$ be a commutative ring with unity.

Let $\struct {R \sqbrk X, \iota, X}$ and $\struct {R \sqbrk Y, \kappa, Y}$ be polynomial rings in one variable over $R$.

Then there exists a unique ring homomorphism $f: R \sqbrk X \to R \sqbrk Y$ such that:

- $f \circ \iota = \kappa$
- $\map f X = Y$

and it is an isomorphism.

## Outline of proof

Using the universal property, we construct ring homomorphisms in both directions, and apply uniqueness to their compositions to show that they are mutual inverses.

## Proof

The existence and uniqueness of $f$ follows from the universal property.

Likewise, there is a unique ring homomorphism $g: R \sqbrk Y \to R \sqbrk X$ such that:

- $g \circ \kappa = \iota$
- $\map g Y = X$

and a unique ring homomorphism $h: R \sqbrk X \to R \sqbrk X$ such that:

- $h \circ \iota = \iota$
- $\map X = X$

By uniqueness and Identity Mapping is Ring Homomorphism, $h = \operatorname{id}$ is the identity mapping on $R \sqbrk X$.

Again by uniqueness and Composition of Ring Homomorphisms is Ring Homomorphism, $g\circ f = \operatorname{id}_{R \sqbrk X}$.

By symmetry, $f \circ g = \operatorname{id}_{R \sqbrk Y}$.

Thus $f$ is an isomorphism.

$\blacksquare$