Definition:Generated Ring Extension/Smallest Subring

Definition
Let $R$ and $S$ be commutative rings with unity.

Let $\phi : R \to S$ be a ring extension of $R$.

Let $T \subseteq S$ be a subset of $S$.

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

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

Also see

 * Equivalence of Definitions of Generated Ring Extension