Definition:Generated Submodule

Definition
Let $R$ be a ring.

Let $M$ be an $R$-module. Let $S\subset M$ be a subset.

Definition 1
The submodule generated by $S$ is the intersection of all submodules of $M$ containing $S$.

Definition 2
The submodule generated by $S$ is the set of all linear combinations of elements of $S$.