Definition:Generated Ring Extension/Smallest Subring

Definition
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 $R \sqbrk T$ $R \sqbrk T$ has no proper subring containing $T$ and $R$.

Also see

 * Equivalence of Definitions of Generated Ring Extension