# Definition:Polynomial Evaluation Homomorphism

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## 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**.