This is somewhat related to [Greg's question](https://mathoverflow.net/questions/2551/why-do-groups-and-abelian-groups-feel-so-different) about groups and abelian groups. Suppose you met someone who was well-acquainted with groups, but who was unwilling to accept rings as a meaningful object of study. How would you describe rings to them in a natural way given that they like talking about groups? 


(Admittedly this is not really the question the title asks.)



What is the "right" definition of a ring?
Well rings are naturally the objects which act on abelian groups - indeed composition always endows the endomorphisms of an abelian group with the structure of a ring. So if one is interested in the endomorphisms of groups one is actually interested in rings.


One can make this analogy more precise especially if one picks a particular ring and looks at the forgetful functor to abelian groups from its module category. This analogy can then be used again for instance to motivate the definition of plethory which are the natural objects which act on rings.


To address Eric's comment about commutative rings there is an analogue in this case of something which was mentioned in the discussion of groups versus abelian groups. Indeed one can obtain commutative rings by considering the identity in an additive symmetric monoidal category. In this case the endomorphisms of the tensor unit are endowed with an abelian group structure via the augmentation over abelian groups and the Eckmann-Hilton argument applied to tensoring endomorphisms and composing endomorphims forces the composition to be abelian. So from this point of view commutative rings are the gadgets which naturally act on the hom-sets of additive symmetric monoidal categories.


Since I mentioned this one can take this slightly further. If one considers such a category together with an autoequivalence (for instance if we take a tensor triangulated category) then one can consider the graded endomorphism ring of the identity. This naturally gives rise to an integer graded ring which is commutative up to some unit which squares to the identity and which has a natural action on the category.


