Universal Property of Polynomial Ring/Free Monoid on Set

Theorem
Let $R, S$ be a commutative and unitary rings, and $\left\{s_j:j\in J\right\}\subseteq S$ be a subset of $S$ indexed by $J$.

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

Let $R \left[{ \left\{X_j:j\in J\right\} } \right]$ be the Ring of Polynomial Forms with coefficients in $R$.

Then there exists a unique homomorphism $\phi: R \left[ { \left\{ X_j:j\in J \right \} } \right] \to S$ extending $\psi$ such that $\phi(X_j) = s_j$ for all $j \in J$.

Remarks

 * The homomorphism $\phi$ is often called evaluation at $\{s_j:j\in J\}$.


 * The requirement that the rings be commutative is vital. A fundamental difference for polynomials over non-commutative rings is additional difficulty identifying polynomial forms and functions using this method.