Definition:Forgetful Functor from Modules to Abelian Groups

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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

Object functor:         sends a left module or right module to its underlying abelian group.
Morphism functor: sends a mapping to itself

Also see