  We review the concept of differentiably simple ring and we give a new proof
of Harper's Theorem on the characterization of Noetherian differentiably simple
rings in positive characteristic. We then study flat families of differentiably
simple rings, or equivalently, finite flat extensions of rings which locally
admit $p$-basis. These extensions are called "Galois extensions of exponent
one". For such an extension $A\subset C$, we introduce an $A$-scheme, called
the "Yuan scheme", which parametrizes subextensions $A\subset B\subset C$ such
that $B\subset C$ is Galois of a fixed rank. So, roughly, the Yuan scheme can
be thought of as a kind of Grassmannian of Galois subextensions. We finally
prove that the Yuan scheme is smooth and compute the dimension of the fibers.
