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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"?

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A module is an abelian group. (It's more useful to think of a module as the analog of a vector space, but with the set of scalars coming from a ring instead of a field. Usually, one arrives at this notion of a module in terms of "the action of a ring on a set" where the set is a module.)

A monoid is a relaxation of the definition of a group. A monoid has an associative operation and a neutral element, but makes no promises about inverses.

I don't see how to express any more of a relation than "all modules are monoids" but only for the dull reason that all (abelian) groups are (abelian) monoids with the added constraint that every element has an inverse.

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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 $F$

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