TITLE: Relations between monoids and modules?
QUESTION [6 upvotes]: What is the relation between monoids and modules? Are they completely different algebraic structures, or is there a kind of inclusion relation like "elements of a module are also elements of a monoid"?

REPLY [4 votes]: There is a chain  of forgetful functors which progressively forgets the various operations in the structure: $$\mathrm{Mod_R}\to\mathrm{Ab}\to\mathrm{AbMon}\to\mathrm{Set}$$
The interesting thing is that you can go in the opposite direction too with free functors $$\mathrm{Set}\to\mathrm{AbMon}\to\mathrm{Ab}\to\mathrm{Mod_R}$$ 
Each forgetful functor $U$ is adjoint to the respective free functor $F$