\begin{definition}[Definition:Generated Ring Extension/Evaluation of Polynomial Ring]
{{:Definition:Generated Ring Extension}}
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 $\Img {\operatorname {ev} }$, the image of $\operatorname {ev}$.
$T$ is said to be a '''generator''' of $R \sqbrk T$.
\end{definition}