Obviously when you can add a second operation satisfying the axioms, but is there an easy characterization of that?

What I mean, let's say I have an abelian group (G,+). When is it possible to keep its structure as a group and define a second operation to have a ring (G,+,×)?

What I want is basically for these to hold:

1. (ab)c=a(bc)

2. a(b+c)=ab+ac

and possibly

3. There exists element e such that for any x in G, ex=xe=x

Ok, how do I actually check these? I mean what I actually want to show is that such an operation is possible, it doesn't necessarily need to "feel like multiplication", so something like 3(2+1) must distribute as 3*2+3*1, but it could equal 17 for all we know. Does the operation even need to be surjective? We don't require inverses so maybe not.

This is just a weird though, I don't know how to approach this, but it just seems like there are some obvious conditions I will feel stupid for not noticing intuitively.