# Equivalence of Definitions of Generated Submodule

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## 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 $S$.

## Proof

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