Definition:Polynomial Ring/Universal Property

Definition
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

 * Definition:Universal Property of Polynomial Algebra
 * Universal Property of Polynomial Ring
 * Equivalence of Definitions of Polynomial Ring