The approach I used seems to work only for totally ordered rings. That's certainly much more general than the approach based on derivatives, but I am rather curious as to whether it's possible to do better. --Dfeuer (talk) 17:50, 12 January 2013 (UTC)
- The more I think about it, the more I doubt it applies without total ordering, since comparison with $0$ is such a key aspect. However, it might be worth considering algebraic generalization, perhaps to rigs. --Dfeuer (talk) 18:33, 12 January 2013 (UTC)