# 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 $\struct {R \sqbrk X, \iota, X}$ be a polynomial ring in one variable over $R$.

Let $s\in S$.

A ring homomorphism $h : R \sqbrk X \to S$ is called an evaluation in $s$ if and only if:

$\map h X = s$
$h \circ \iota = \kappa$

where $\circ$ denotes composition of mappings.

### Multiple Indeterminates

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 $\family {s_i}_{i \mathop \in I}$ if and only if:

$\forall r \in R : \map g r = \map \kappa r$
$\forall j \in J : \map g {X_j} = s_j$

## Also known as

The evaluation homomorphism is also known as substitution homomorphism.