[gpm]

You need slightly more than being a ring. It's possible to make a ring (even a field) over R^n provided you don't care about interactions with scalars. For example: Just take any one-to-one map from R^n to R and then apply the operations in R before mapping the results back. It won't make any geometric sense, but it will be a ring.

I've been blanking on what exactly the interactions with scalars that we need to preserve are...