Definition:Generated Ring Extension/Evaluation of Polynomial Ring
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Definition
Let $S$ be a commutative rings with unity.
Let $R$ be a subring of $S$ with unity such that the unity of $R$ is the unity of $S$.
That is, $S$ is a ring extension of $R$.
Let $T \subseteq S$ be a subset of $S$.
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 $R \sqbrk T$.
Also see
- Universal Property of Polynomial Ring, for a proof of the existence of a unique polynomial evaluation homomorphism at $T$.
Sources
- Weisstein, Eric W. "Extension Ring." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExtensionRing.html