I'm trying to get comfortable with concepts from synthetic differential geometry. I was wondering whether the structure of a Weil algebra is somehow related to the general notion of an infinitesimal extension of a ring, which is a (regular?) epi $\hat R\twoheadrightarrow R$ with nilpotent kernel.
The structure of a Weil algebra is still mostly opaque to me, but this general notion of an infinitesimal extension looks very intuitive, so I'm hoping for a simple link..
Weil algebras over $k$ are infinitesimal extensions of $k$.