Definition:Generated Submodule/Unitary

Definition
Let $R$ be a ring with unity.

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

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

Also see

 * Equivalence of Definitions of Generated Submodule over Ring with Unity, which shows that the set defined here is a submodule of $M$.