29.
In this paper restricted differential operator rings are studied. A restricted differential operator ring is an extension
of a
k-algebra
R by the restricted enveloping algebra of a restricted Lie algebra g which acts on
R. This is an example of a smash product
R #
H where
H=
u (g). We actually deal with a more general twisted construction denoted by
R * g where the restricted Lie algebra g is not necessarily embedded isomorphically in
R * g. Assume that g is finite dimensional abelian. The principal result obtained is Incomparability, which states that prime
ideals
P
1 ⊆
P
2 ⊂
R * g have different intersections with
R. We also study minimal prime ideals of
R * g when
R is g-prime, showing that the minimal primes are precisely those having trivial intersection with
R, 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.
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