2. Preliminaries for S = R[y\r,8].
The aim of this section is to sketch some basic background results on skew
derivations and skew polynomial rings which will be used throughout the paper.
In particular, we discuss the (/-skew derivations and g-skew polynomial rings
studied in [13], and we outline one class of examples.
Throughout this section R will denote a noetherian ring.
2.0. (i) We will rely on basic results from the theory of noetherian rings.
Included are properties of right Krull dimension and classical Krull dimension,
Goldie's Theorem, and elementary facts concerning localization at Ore sets.
The reader is referred to [16], [21], or [34] for more information.
(ii) We denote the right Krull dimension of R by rKdimJ?, the classical
Krull dimension of R by clKdimi?, the set of prime ideals of R, by speci?, and
the set of maximal ideals of R by maxi?. Endow speci? with the Jacobson
(Zariski) topology, and each subset with the subspace topology. If Q is a prime
ideal of R which is the annihilator of a submodule of an i?-module M , then Q
is said to be an annihilator prime of M. Now suppose that a ring S contains
R as a subring. If P is a prime ideal of S and Q is a prime ideal of R then
P is said to lie over Q provided Q is minimal over P 0 R. If in addition Q is
an annihilator prime of the right i?-module
(S/P)R,
we shall say that P lies
directly over Q (see (5.1), (5.2)). Finally, " c " will denote proper containment.
We next briefly summarize some basic information about skew polynomial
rings, much of which is found in greater detail in [13, Sections 1 and 2].
2.1 . (i) Let r denote a ring endomorphism of R. A (left) r-derivation 8 of R
is an additive map from R to itself such that 8(ab) = r(a)6(b) + 8(a)b for all
a, 6 G R- The pair (r, 8) is referred to as a (left) skew derivation.
(ii) Let X denote a subset of i?, and let E denote a set of functions from
R to itself. If &(X) C X for all a G E, we say that X is Testable. A E-ideal
of R is an ideal which is E-stable. A proper E-ideal K of R is Yi-prime if for
each pair of E-ideals / , J with IJ C K it follows that / C K or J C K. We
refer to the collection of E-prime ideals of R as spec
s
i?, and we endow it with
the obvious generalization of the Jacobson topology. The ring R is said to
be E-prime provided (0) is a E-prime ideal, and when R is nonzero it is said
to be H-simple provided (0) and R are the only E-ideals of R. Let r be an
6
Previous Page Next Page