Definition:Generator of Module

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

Let $S \subseteq G$.

The submodule generated by $S$ is the smallest submodule $H$ of $G$ containing $S$.

In this context, we say that:
 * $S$ a generating system for $H$ (over $R$)
 * $S$ a generating set for $H$ (over $R$)
 * $S$ generates $H$
 * $S$ is a set of generators for $H$ (over $R$)
 * $S$ is a generator for $H$ (over $R$)

If $R$ is a field, then: or
 * $S$ is a spanning set for $H$ (over $R$)
 * $S$ spans $H$.

This definition also applies when $G$ is a vector space.