Universal Property of Polynomial Ring/Free Monoid on Set

Theorem
Let $R, S$ be a commutative and unitary rings, and $s_1, \ldots, s_n \in S$.

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

Let $R \left[{X_1, \ldots, X_n}\right]$ be the Ring of Polynomial Forms in $n$ indeterminates with coefficients in $R$.

Then there exists a unique homomorphism $\phi: R \left[{X_1, \ldots, X_n}\right] \to S$ extending $\psi$ such that $\phi(X_j) = s_j$ for $j = 1, \ldots, n$.

Remarks

 * The homomorphism $\phi$ is often called evaluation at $s$.


 * 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.