\begin{definition}[Definition:Generated Submodule/Definition 1]
Let $R$ be a ring.
Let $M = \struct {G, +, \circ}_R$ be an $R$-module.
Let $S \subset M$ be a subset of $M$.
The '''submodule generated by $S$''' is the intersection of all submodules of $M$ containing $S$.
\end{definition}