Definition:Polynomial Evaluation Homomorphism

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

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

Single indeterminate
Let $(R[X], \iota, X)$ be a polynomial ring in one variable over $R$.

Let $s\in S$.

A ring homomorphism $h : R[X] \to S$ is called evaluation in $s$ :
 * $h(X) = s$
 * $h \circ \iota = \kappa$

where $\circ$ denotes composition of mappings.

Multiple indeterminates
Let $R, S$ be commutative rings with unity.

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

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

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

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

Also known as
The evaluation homomorphism is also known as substitution homomorphism.

Also see

 * Definition:Universal Property of Polynomial Ring