\begin{definition}[Definition:Forgetful Functor from Modules to Abelian Groups]
Let $R$ be a ring.
Let $\mathbf C$ be the category of left modules or category of left modules over $R$.
Let $\mathbf{Ab}$ be the category of abelian groups.
The '''forgetful functor''' $\mathbf C \to \mathbf{Ab}$ is the covariant functor with
{{DefineFunctor
| ob = sends a left module or right module to its underlying abelian group.
| mor = sends a mapping to itself
}}
\end{definition}