Here is some more support for rings as objects that act on abelian groups.

Someone well-acquainted with groups would likely know that representations of groups are important to their study.  Given a group G, a representation of G over a field k is an action of G on a k-vector space V as a group of linear automorphisms.  Familiarity with rings allows us to realize that this is the same as a *ring homomorphism* from the [group ring][1] kG into End<sub>k</sub>(V).  Then one can immediately begin to investigate group actions by asking questions about the structure of the group ring kG.  In fact, one can even show that the category of G-modules (representations of G) is equivalent to the category of (say left) modules over the ring kG.

In this way, rings generalize the notion of groups acting on objects with additive structure.  In this perspective, rings are important *because they act on modules.*

  [1]: http://en.wikipedia.org/wiki/Group_ring