Universal Property of Polynomial Ring

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

The constructions of a polynomial ring $R[X]$ on $R$ satisfy the following universal property:


 * For every:
 * commutative ring with unity $S$
 * ring homomorphism $g : R \to S$
 * element $s\in S$
 * there exists a unique ring homomorphism $h : R[X] \to S$ with $h(X) = s$ and $h \circ f = g$
 * where:
 * $X$ is the variable of $R[X]$
 * $f$ is the embedding of $R$ in $R[X]$.

Also see

 * Equivalence of Definitions of Polynomial Ring