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 $S$ is a generating set for $H$, or that $S$ generates $H$.

If we have $R$ as a field, we instead say S is a spanning set or that $S$ spans $H$.

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