I am confused by properties of $R$, the geometric line in synthetic differential geometry. In the book Synthetic Differential Geometry by Kock, he assumes $R$ is only a commutative ring. However, in these notes by Mike Shulman, it is said that $R$ is a field in the following constructive sense: $$x\neq 0\implies x\text{ is invertible}.$$
This seems to be an assumption which Kock does not make: he explicitly states that we must sometimes assume $R$ is a $\mathbb Q$-algebra, for instance in order to prove $D$ is not an ideal.
Why should or shouldn't one make this assumption?