131 Differential rings

Consider a set $ℛ$ of complex functions on an open set $\mathrm{\Omega }$ such that

(R) $ℛ$ is a ring, so that $f,g\in ℛ$ and $\lambda ,\mu \in 𝐂$ imply $fg\in ℛ$ and $\lambda f+\mu g\in ℛ$;

(C) multiplication is commutative $f(s)g(s)=g(s)f(s)$;

(ID2) $f(s)g(s)=0$ for all $s\in \mathrm{\Omega }$ implies $f(s)=0$ or $g(s)=0$ on $\mathrm{\Omega }$;

(Diff) For all $f(s)\in ℛ$ the derivative $f^{\prime }(s)$ also belongs to $ℛ$, and satisfies Leibniz’s rule ${(fg)}^{\prime }=f^{\prime }g+f{g}^{\prime }$.

We then form

$$ℱ=\{f=g/h:g,h\in ℛ;h\ne 0\}$$

with the usual multiplication and

$$f^{\prime }=\frac{g^{\prime }h-g{h}^{\prime }}{h^2}.$$

This is called the field generated by $ℛ$.