How can *$R$ be a field in Synthetic Differential Geometry or Smooth Infinitesimal analysis?

Question

In many sources it was written that *$R$ is ring and *$R$ can't be a field because it's not possible that $d^2=0$ in field.

But in some sources it was written that *$R$ is a field.

How can *$R$ be a field in Synthetic Differential Geometry or Smooth Infinitesimal Analysis?

Thanks.

Answer 1

Since SDG uses intuitionistic logic, it is important to specify how exactly you define a field. If you want the property that if $x\not=0$ then $x$ is invertible, then this still holds. Nilsquare infinitesimals do not provide a counterexample to this, since they are not provably nonzero.