# Equivalence of Definitions of Generated Submodule

## Theorem

Let $R$ be a ring.

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

Let $S\subset M$ be a subset.

The following definitions of the concept of Generated Submodule are equivalent:

### 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 $S$.