  We consider a circle of ideas involving differential algebra, local
Noetherian rings, and their generic formal fibers.
  Connecting these ideas gives rise to what we term a "twisted" subring $R$ of
a ring $S$. Each such subring $R$ arises as a pullback of a derivation taking
values in an $S$-module $K$. The twisting relationship proves to be a kind of
inversion of Nagata idealization: whereas idealization extends $S$ to the
larger ring $S \star K$, twisting produces a subring of $S$ which behaves much
like the ring $S \star K$.
  The rings produced in this manner exhibit pathological features, such as
failing to have finite normalization, but in spite of this they are quite
tractable and conceptually (if not practically) easy to locate, and in this way
provide a rich but manageable source of non-standard Noetherian rings.
  The theory developed to support this construction involves extensive use of
both Noetherian and non-Noetherian commutative ring theory, as well as
differential algebra.
