Definition:Generated Submodule/Definition 1

Definition
Let $R$ be a ring.

Let $M$ be an $R$-module.

Let $S\subset M$ be a subset.

The submodule generated by $S$ is the intersection of all submodules of $M$ containing $S$.

Also see

 * Intersection of Submodules is Submodule
 * Definition:Lattice of Submodules