Definition:Polynomial Ring/Universal Property

Definition
Let $R$ be a commutative ring with unity.

A polynomial ring over $R$ is an ordered triple $(S, f, X)$ where:
 * $S$ is a commutative ring with unity.
 * $f : R \to S$ is a unital ring homomorphism
 * $X$ is an element of $S$

that satisfy the following universal property:
 * For every ordered triple $(A, g, a)$ where:
 * $A$ is a commutative ring with unity
 * $g : R \to A$ is a unital ring homomorphism
 * $a$ is an element of $A$
 * there exists a unique evaluation homomorphism $h : S\to A$ in $a$.

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

Also see

 * Universal Property of Polynomial Ring
 * Equivalence of Definitions of Polynomial Ring