\begin{definition}[Definition:Generated Ring Extension/Smallest Subring]
{{:Definition:Generated Ring Extension}}
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 $R \sqbrk T$ {{iff}} $R \sqbrk T$ has no proper subring containing $T$ and $R$.
\end{definition}