  We revisit the concept of special algebras, also known as \textit{purely
inseparable ring extensions}. This concept extends the notion of purely
inseparable field extensions to the more general context of extensions of
commutative rings. We use differential operators methods to provide a
characterization for a ring extension to be purely inseparable in terms of a
condition on certain modules of differential operators associated to the ring
extension. This approach is also used to recover an already known
characterization involving the modules of principal parts.
  Next, given a purely inseparable ring extension $A\subset C$, we aim to
understand which intermediate rings $A\subset B\subset C$ satisfy the property
that both $A\subset B$ and $B\subset C$ are both flat extensions by considering
only the subalgebra $\operatorname{End}_B(C)$ of $\operatorname{End}_A(C)$. To
achieve this, we prove a generalization of the Jacobson-Bourbaki theorem on
Galois correspondence for field extensions to the setting of commutative ring
extensions with homeomorphic spectra.
  Finally, given a tower of ring extensions $A\subset B\subset C$, we consider
the question of whether the fact that two of the three extensions $A\subset C$,
$A\subset B$, and $B\subset C$ are purely inseparable implies that the third
one is also purely inseparable.
