TITLE: Why is there no Cayley's Theorem for rings?
QUESTION [58 upvotes]: Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a technical theorem as a glorious wellspring of intuition- something, at least from my perspective, that rings are missing; and I want to know why.
Certainly the axiom system is more complicated- so there is no way you're going to get as simple a characterisation as you do with groups- but surely there must be some sort of universal object for rings of a given cardinality, analogous to the symmetric group in group theory. I would be surprised if it was a ring- the multiplicative and additive properties of a ring could be changed (somewhat) independently of one another- but perhaps a fibration of automorphisms over a group? If so is there a natural(ish) way of interpreting it?
Perhaps it's possible for a certain subclass of rings, perhaps it's possible but useless, perhaps it's impossible for specific reasons, in which case: the more specific the better. 
Edit: So Jack's answer seems to have covered it (and quickly!): endomorphisms of abelian groups is nice! But can we do better? Is there a chance that 'abelian' can be unwound to the extent we can make this about sets again- or is that too much to hope for?

REPLY [45 votes]: You can think of Cayley's theorem as the special case of Yoneda's lemma, where the category has only one object. If you take the additive version of Yoneda's lemma, and you plug in an additive category with one object then you get the desired statement for rings that's in Jack's answer. Viewing Cayley's theorem like this let's you generalize it to many other structures then groups.