Ring Isomorphic to Polynomial Ring is Polynomial Ring

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Theorem

One Variable

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 $R$.


Multiple Variables