**"Frac" and "Const" commute**: A differential ring is simply a pair $(R,\partial)$ where $R$ is a ring and $\partial$ is a derivation in $R$ (i.e. $\partial\in End_{\mathbb{Z}}(R)$ follows identically Liebniz rule $\partial(ab)=\partial(a)b+a\partial(b)$). The constants of $R$, $\ker(\partial)$ form a subring $Const(R)\subset R$ called _subring of constants_.

If $R$ is a commutative domain (i.e. without zero divisors), it is standard to consider the field of fractions $Frac(R)$ and to extend $\partial$ by the formula of calculus $\partial(1/g)=-\partial(g)/g^2$.     

**False belief**: $Const(Frac(R))=Frac(Const(R))$, in other words, every constant in $Frac(R)$ is of the form $\alpha/\beta$, where $\alpha,\beta\in Const(R)$.  

Until yesterday morning, I postponed to prove this (false) lemma (thinking it was an easy exercise). Then, preparing a talk, 
I could not prove this and searched for a counterexample. Finally I found a simple one in MSE (see below).  

[See there for a discussion and counterexamples](https://math.stackexchange.com/questions/2051634).