Originally Posted by

**topsquark** This is more of a lark than anything else.

I've been working with rings and I had a thought. Rings have two binary operations on the elements. What if we have a multiplicative group and wanted to impose a ring structure on it? My example is say we have $\displaystyle D_6 : \{r^3 = s^2 = 1,~rs = s r^{-1} \}$. Is there a process to write an additive binary operation for this? (I doubt it would have a geometric meaning in this case.) I can construct one but there are a number of arbitrary choices to make. Is there a "standard" procedure to do this in a systematic way (ie. get rid of the arbitrary choices) that would make the system a ring? I suspect that the different addition tables I can write are the same up to a relabeling of the elements and I haven't actually checked to see if the addition tables are associative. I don't know how to check to see if the elements are a group under the addition operation in general.