Definition:Generated Ring Extension

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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 $S$.

Definition 1

The ring extension $R \sqbrk T$ generated by $T$ is the smallest subring of $S$ containing $T$ and $R$, that is, the intersection of all subrings of $S$ containing $T$ and $R$.

Thus $T$ is a generator of $R \sqbrk T$ if and only if $R \sqbrk T$ has no proper subring containing $T$ and $R$.

Definition 2

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 $R \sqbrk T$.

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