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
- Results about generators of modules can be found here.
Sources
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach