Lecture notes- Applied Stability Theory (Math 414) Autumn 2003.

Sometimes the derivation is denoted by a→ a′. A ring equipped with a derivation is called a differential ring. The notions, differential subring, differential ring extension and homomorphism between differential rings, are clear. If (R, ∂) is a differential ring then the set of constants is by definition {r ∈ R : ∂(r) = 0}, and is easily seen to be a (differential) subring. ∂ means the k-fold iteration of ∂ and ∂ is taken to be the identity mapping. The basic examples of differential rings are rings (or fields) of functions: for example (i) C(z) the field of rational functions over C in the single indeterminate z, with ∂(f) = df/dz, (ii) C(z, e) with ∂ as in (i). (Note that this field IS closed under ∂.)