Definition:Polynomial Ring/Universal Property

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Let $R$ be a commutative ring with unity.

A polynomial ring over $R$ is a pointed $R$-algebra $(S, \iota, X)$ that satisfies the following universal property:

For every pointed $R$-algebra $(A, \kappa, a)$ there exists a unique pointed algebra homomorphism $h : S\to A$, called evaluation homomorphism.

This is known as the universal property of a polynomial ring.

Also see