Definition:Generated Submodule/Definition 1

Definition
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 $S$.

Also see

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