# Universal Property of Polynomial Ring

## Theorem

Let $R$ be a commutative ring with unity.

The different definitions of a polynomial ring $\struct {\map R x, \iota, x}$ on $R$ satisfy the universal property of a polynomial ring:

### Construction using Free Monoid on Set

Let $R, S$ be commutative and unitary rings.

Let $\left\langle{s_j}\right\rangle_{j \mathop \in J}$ be an indexed family of elements of $S$.

Let $\psi: R \to S$ be a ring homomorphism.

Let $R \left[{\left\{{X_j: j \in J}\right\}}\right]$ be a polynomial ring.

Then there exists a unique evaluation homomorphism $\phi: R \left[{\left\{{X_j: j \in J}\right\}}\right] \to S$ at $\langle s_j\rangle_{j \in J}$ extending $\psi$.