1. Introduction.

The study of ideals and representations of an algebra S is often based on

the dual techniques of induction and restriction between S and a suitable sub-

algebra R. In some of the most extensively studied cases, S is equal to (or is

a factor ring of) a skew polynomial ring R[y; T,S]. In fact, S is often obtained

through an iterated skew polynomial construction ^[y1][y2; ^2? ^2] * * * [l/n]

Tn

^n]

over a base field A:. Classical instances include Weyl algebras, enveloping al-

gebras of solvable Lie algebras, group algebras of polycyclic groups, and the

enveloping algebra U(s\2(k)). Examples currently under scrutiny are q-Weyl

algebras, enveloping algebras of solvable Lie superalgebras, the ^-enveloping al-

gebra Uq(s\2(k)), and certain other quantum groups (e.g., [13], [32], [38]). In

the "unmixed" cases - those of skew polynomial rings R[y\ r, 6] where either r

or 6 is trivial - a vast existing literature provides a thorough understanding of

most of the classical iterated skew polynomial rings. Many of these methods,

however, cannot be extended to the case where both r and 6 are nontrivial, as

easy examples show. Thus different tools are required for investigations of the

more recent instances of iterated skew polynomial rings.

Our primary aim is to introduce new methods for analyzing the structure

of noetherian skew polynomial rings and, in particular, to develop means to

find and classify the prime ideals in such rings. We obtain the most precise

results for the q-skew extensions introduced in [13], namely skew polynomial

rings R[y, r, 6] in which

T~16T

= q8 for some nonzero scalar q. In this setting -

which includes for example the skew polynomial extensions appearing in g-Weyl

algebras, enveloping algebras of solvable Lie superalgebras, and coordinate rings

of quantum matrices - we present a complete description of the prime ideals.

Next, we test our analysis of the 5-skew case by considering in more detail some

Received by the editor October 22, 1991, and in revised form December 28, 1992.

The research of the first author was partially supported by a grant from the National

Science Foundation.

Most of the second author's research for this paper was supported by a National

Science Foundation postdoctoral research fellowship. Part of the research was done

during October and November of 1990 while visting the Weizmann Institute of Science;

the hospitality of this institute is gratefully acknowledged.

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