6 ms·

I don't see how you would define linearity in a ring (I don't mean a module, just a ring). I.e. D(af) = aD(f) doesn't make sense if you don't have scalar multip

• gizmo686 2y ago
  The minimum would be to ask for D(a+b) = D(a) + D(b). EDIT: Actually, according to wikipedia, that is exactly what is done in differential algebra https://en.wikipedia.org/wiki/Differential_algebra https://en.wikipedia.org/wiki/Differential_algebra
• JadeNB 2y ago
  > I don't see how you would define linearity in a ring (I don't mean a module, just a ring). I.e. D(af) = aD(f) doesn't make sense if you don't have scalar multiplication. Every ring is a module over itself. But you wouldn't want the definition you propose; instead, you'd want `D(ab) = aD(b) + D(a)b`. If you really like some sort of linearity to be present, you could observe that this property forces every derivation to be linear as a transformation of `R_0`-modules, where `R_0` is the subring `ker(D)` of "constants".