# Definition:Generator of Module

## Definition

Let $R$ be a ring.

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

Let $S \subseteq M$ be a subset.

### Definition 1

$S$ is a **generator of $M$** if and only if $M$ is the submodule generated by $S$.

### Definition 2

$S$ is a **generator of $M$** if and only if $M$ has no proper submodule containing $S$.

## Generator of Unitary Module

Let $R$ be a ring with unity.

Let $M$ be a unitary $R$-module.

Let $S \subseteq M$ be a subset.

$S$ is a **generator of $M$** if and only if every element of $M$ is a linear combination of elements of $S$.

## Generated Submodule

Let $R$ be a ring.

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

Let $S \subseteq M$ be a subset.

### $R$-module

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

### Unitary $R$-Module

Let $R$ be a ring with unity.

Let $M$ be a unitary $R$-module.

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

## Also known as

A **generator** of a module is also known as a **spanning set**.

Some sources refer to a **generator for** rather than **generator of**. The two terms mean the same.

It can also be said that $S$ **generates $M$ (over $R$)**.

Other terms for $S$ are:

- A
**generating set of $M$ (over $R$)** - A
**generating system of $M$ (over $R$)**

Some sources refer to such an $S$ as a **set of generators of $M$ over $R$** but this terminology is misleading, as it can be interpreted to mean that each of the elements of $S$ is itself a **generator** of $M$ independently of the other elements.

## Also see

- Equivalence of Definitions of Generator of Module
- Equivalence of Definitions of Generator of Unitary Module

## Sources

- 1970: George Arfken:
*Mathematical Methods for Physicists*(2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach