What a ring is, really, is just a collection of homomorphisms from a commutative group to itself.
This is true in the same way that sentences like
What a monoid is, really, is just a collection of functions from a set to itself.
What a group is, really, is just a collection of bijections from a set to itself.
are true --- which is to say, there's a relevant Cayley/Yoneda theorem. In this case it says the category of rings (and ring homomorphisms) is equivalent to the category
- whose objects are pairs (X, M) where X is a commutative group and M is a collection of group homomorphisms X → X, such that for any f, g ∈ M we have also that f o g ∈ M.
- whose arrows (X, M) → (X', M') → are maps f : M → M' such that f preserves composition and also "addition" of arrows in M, which is defined as (h + k)(x) = h(x) + k(x) for h, k ∈ M and where the + on the rhs is the group operation of X. To say that f preserves addition is to say that f(h + k) = fh + fk.
So the thing I wonder is, is there a nice way I can I avoid "inventing" and stipulating the preservation of addition? Is there a categorical crank I can turn such that I just stick the category Ab into the machine and out pops Ring, and stick in Sets and get out Mon? What happens if I put in Mon, ComMon, Ring?
Whoa also Frucht's Theorem! Every group arises as the set of all automorphisms of some graph. That's much stronger than every group arises as a (composition-closed) subset of the automorphisms of some set.