Earlier today I had a conversation with a friend about ways of putting topologies on sets of first-order structures; we wound up talking about reducts and expansions from a topological point of view, and this question arose as a special case.

The question has a specific part, and a less specific part. Hopefully these are both appropriate questions for MO; not being an algebraist, I'm not certain these aren't trivial. (Also, in this question I flagrantly abuse notation in the hopes of making things more readable.)

For $G$ a group with underlying set $X$, let $\mathcal{R}_G$ denote the set of possible ring structures with additive group $G$. We can give $\mathcal{R}_G$ a topology by taking as a sub-basis all sets of the form $\lbrace R: a\times_Rb=c\rbrace$ for $a, b, c\in G$. This is (when nonempty) a totally disconnected space, since any two distinct ring structures with the same additive group must disagree on the product of some pair of elements. The automorphism group $Aut(G)$ acts on $\mathcal{R}_G$ by homeomorphisms in a natural way, and for each $R\in\mathcal{R}_G$ the group $Aut(R)$ acts on $\mathcal{R}_G$ by homeomorphisms fixing the point $R$; but in general, I have difficulty saying much of anything about this space. My first question is: what can be said about the spaces $\mathcal{R}_G$ or the map $G\mapsto\mathcal{R}_G$, and what is a good source about this?

My vaguer question is this: what is the "right" way of looking at all the compatible ring structures on a given group? I get the sense, just from playing around with it, that the topology $\mathcal{R}_G$ is too coarse an object (in a completely non-rigorous sense of the word) to get much information out of. Presumably there is a better way of looking at the same set of objects; what is it?