Definition:Generated Ring Extension/Smallest Subring

Definition
Let $E / F$ be a field extension.

Let $S \subset E$ be a subset of $E$.

The field extension $F \sqbrk S$ generated by $S$ is the smallest subfield extension of $E$ containing $S$, that is, the intersection of all subfields of $E$ containing $S$ and $F$.

Thus $S$ is a generator of $F \sqbrk S$ $E$ has no proper subfield extension containing $S$.

Also see

 * Equivalence of Definitions of Generated Ring Extension