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 kG into End_k(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. In this vein, every ring can be realized as an endomorphism ring of a module: for a ring R, the right module R_R satisfies R ≅ End(R_R). (In analogy with the terminology for group representations, R_R is sometimes referred to as the regular representation of R.)

Here is some more support for rings as objects that act on abelian groups, as already mentioned by Greg Stevenson.

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 kG into End_k(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.

From this perspective, rings are important because they act on modules. In this vein, every ring can be realized as an endomorphism ring of a module: for a ring R, the right module R_R satisfies R ≅ End(R_R). (In analogy with the terminology for group representations, R_R is sometimes referred to as the regular representation of R.)

To go one step further, the endomorphism ring of an object in any abelian (or even preadditive) category is a ring. (Though from Greg's post, it sounds as if one can go even further than this!) So we see that rings greatly generalize the notion of groups acting on objects with additive structure.

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 kG into End_k(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.

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 kG into End_k(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. In this vein, every ring can be realized as an endomorphism ring of a module: for a ring R, the right module R_R satisfies R ≅ End(R_R). (In analogy with the terminology for group representations, R_R is sometimes referred to as the regular representation of R.)