Definition:Polynomial Evaluation Homomorphism

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

Let $\left(s_j\right)_{j \in J}$ be a family of elements of $S$.

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

Let $R \left[{\left\{{X_j: j \in J}\right\}}\right]$ be a ring of polynomial forms over $R$.

A homomorphism $\phi: R \left[{\left\{{X_j: j \in J}\right\}}\right] \to S$ is called evaluation at $\left(s_j\right)_{j \in J}$ if it extends $\psi$ and sends $X_j$ to $s_j$ for all $j\in J$.

Also see

 * Evaluation Homomorphism, where it is proven that an evaluation morphism exists and is unique, in the case of commutative and unitary rings.