\begin{definition}[Definition:Generator of Algebra]
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$.
Let $S \subseteq A_R$ be a subset of $A_R$.
The '''subalgebra generated by $S$''' is the smallest subalgebra $B_R$ of $A_R$ which contains $S$.
\end{definition}