# Definition:Polynomial Evaluation Homomorphism/Single Indeterminate

## Definition

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

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

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.

## Also known as

The evaluation homomorphism is also known as substitution homomorphism.