\begin{definition}[Definition:Polynomial Evaluation Homomorphism/Multiple Indeterminates]
Let $R, S$ be commutative rings with unity.
Let $\kappa : R \to S$ be a unital ring homomorphism.
Let $\family {s_i}_{i \mathop \in I}$ be an indexed family of elements of $S$.
Let $R \sqbrk {\set {X_i: i \mathop \in I} }$ be a polynomial ring over $R$.
A ring homomorphism $g: R \sqbrk {\set {X_i: i \mathop \in I} } \to S$ is called an '''evaluation at $\family {s_i}_{i \mathop \in I}$''' {{iff}}:
:$\forall r \in R : \map g r = \map \kappa r$
:$\forall j \in J : \map g {X_j} = s_j$
\end{definition}