Model theory of fields with free operators in positive characteristic

We give algebraic conditions for a finite commutative algebra B over a field of positive characteristic, which are equivalent to the companion-ability of the theory of fields with “B-operators” (i.e., the operators comingfrom homomorphisms into tensor products with B). We show that, in themost interesting case of a local B, these model companions admit quantifierelimination in the “smallest possible” language, and they are strictly stable.We also describe the forking relation there.