Definition:Polynomial Evaluation Homomorphism

Definition
Let $R, S$ be commutative rings with unity.

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

Let $\left(s_i\right)_{i \in I}$ be a family of elements of $S$.

Let $R \left[{\left\{{X_i: i \in I}\right\}}\right]$ be a polynomial ring over $R$.

A ring homomorphism $g: R \left[{\left\{{X_i: i \in I}\right\}}\right] \to S$ is called evaluation at $\left(s_i\right)_{i \in I}$ if:
 * $\forall r\in R : g(r) = f(r)$
 * $\forall j\in J : g(X_j) = s_j$

Also see

 * Evaluation Homomorphism, where it is proved that an evaluation morphism exists and is unique