Definition:Generated Ring Extension/Evaluation of Polynomial Ring

Definition
Let $R \sqbrk {\set {X_t} }$ be the polynomial ring in $T$ variables $X_t$.

Let $\operatorname {ev} : R \sqbrk {\set {X_t} } \to S$ be the evaluation homomorphism associated with the inclusion $T \hookrightarrow S$.

The ring extension $R \sqbrk T$ generated by $T$ is $\map {\operatorname {Img}} {\operatorname {ev}}$, the image of $\operatorname {ev}$.

$T$ is said to be a generator of $R \sqbrk T$.

Also see

 * Definition:Polynomial Evaluation Homomorphism


 * Universal Property of Polynomial Ring, for a proof of the existence of a unique polynomial evaluation homomorphism at $T$.


 * Equivalence of Definitions of Generated Ring Extension