\begin{definition}[Definition:Polynomial Ring/Universal Property]
Let $R$ be a commutative ring with unity.
A '''polynomial ring over $R$''' is a pointed $R$-algebra $(S, \iota, X)$ that satisfies the following universal property:
:For every pointed $R$-algebra $(A, \kappa, a)$ there exists a unique pointed algebra homomorphism $h : S\to A$, called evaluation homomorphism.
This is known as the '''universal property of a polynomial ring'''.
\end{definition}