Ring Isomorphic to Polynomial Ring is Polynomial Ring/One Variable
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Theorem
Let $R$ be a commutative ring with unity.
Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$.
Let $\iota : R \to R \sqbrk X$ denote the canonical embedding.
Let $S$ be a commutative ring with unity and $f: R \sqbrk X \to S$ be a ring isomorphism.
Then $\struct {S, f \circ \iota, \map f X}$ is a polynomial ring in one variable $\map f X$ over $R$.
Proof
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