  Let $L/K$ be a finite, totally ramified $p$-extension of complete local
fields with residue fields of characteristic $p > 0$, and let $A$ be a
$K$-algebra acting on $L$. We define the concept of an $A$-scaffold on $L$,
thereby extending and refining the notion of a Galois scaffold considered in
several previous papers, where $L/K$ was Galois and $A=K[G]$ for
$G=\mathrm{Gal}(L/K)$. When a suitable $A$-scaffold exists, we show how to
answer questions generalizing those of classical integral Galois module theory.
We give a necessary and sufficient condition, involving only numerical
parameters, for a given fractional ideal to be free over its associated order
in $A$. We also show how to determine the number of generators required when it
is not free, along with the embedding dimension of the associated order. In the
Galois case, the numerical parameters are the ramification breaks associated
with $L/K$. We apply these results to biquadratic Galois extensions in
characteristic 2, and to totally and weakly ramified Galois $p$-extensions in
characteristic $p$. We also apply our results to the non-classical situation
where $L/K$ is a finite primitive purely inseparable extension of arbitrary
exponent that is acted on, via a higher derivation (but in many different
ways), by the divided power $K$-Hopf algebra.
