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 $M$ every element of $M$ is a linear combination of elements of $S$.

Definition 2
$S$ is a generator of $M$ $M$ has no proper submodule containing $S$.

Definition 3
$S$ is a generator of $M$ $M$ is the submodule generated by $S$.


 * $S$ is a generator of $M$ (over $R$)
 * $S$ generates $M$ (over $R$).

All are equivalent.

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

Other terms for $S$ are:
 * A generating set of $H$ (over $R$)
 * A generating system of $H$ (over $R$)

Some sources refer to such an $S$ as a set of generators of $H$ 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 $H$ independently of the other elements.

Also see

 * Equivalence of Definitions of Generator of Module