Definition:Generator of Module

Definition
Let $G$ be an $R$-module.

Let $S \subseteq G$.

Let $H$ be the smallest submodule $H$ of $G$ containing $S$.

Then:
 * $S$ is a generator of $H$ (over $R$)
 * $H$ is the submodule generated by $S$
 * $S$ generates $H$ (over $R$).

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

 * Definition:Generator