What is meant by the underlying set of an $R$-module, say $M$? Does it mean $M$ without the module structure, and just looked at as a set, and what does it mean to drop its module structure? Or does it mean the set used to generate $M$, and what does that look like?
There reason I ask is because I'm asked to write down what the forgetful functor $G: R\text{-mods} \to Sets$, defined by sending $R$-mods to their underlying sets, does to morphisms. Any help appreciated.