  We give algebraic conditions about a finite algebra $B$ over a perfect field
of positive characteristic, which are equivalent to the companionability of the
theory of fields with "$B$-operators" (i.e. the operators coming from
homomorphisms into tensor products with $B$). We show that, in the most
interesting case of a local $B$, these model companions admit quantifier
elimination in the "smallest possible" language and they are strictly stable.
We also describe the forking relation there.
