Prime ideals in restricted differential operator rings

In this paper restricted differential operator rings are studied. A restricted differential operator ring is an extension of ak-algebraR by the restricted enveloping algebra of a restricted Lie algebra g which acts onR. This is an example of a smash productR #H whereH=u (g). We actually deal with a more general twisted construction denoted byR * g where the restricted Lie algebra g is not necessarily embedded isomorphically inR * g. Assume that g is finite dimensional abelian. The principal result obtained is Incomparability, which states that prime idealsP ₁ ⊆P ₂ ⊂R * g have different intersections withR. We also study minimal prime ideals ofR * g whenR is g-prime, showing that the minimal primes are precisely those having trivial intersection withR, that these primes are finite in number, and their intersection is a nilpotent ideal. Prime and primitive ranks are considered as an application of the foregoing results.