# Prime Ideals in Skew and \(q\)-Skew Polynomial Rings

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*K.R. Goodearl; E.S. Letzter*

There has been continued interest in skew polynomial rings and related constructions since Ore's initial studies in the 1930s. New examples not covered by previous analyses have arisen in the current study of quantum groups. The aim of this work is to introduce and develop new techniques for understanding the prime ideals in skew polynomial rings \(S=R[y;\tau , \delta ]\), for automorphisms \(\tau\) and \(\tau\)-derivations \(\delta\) of a noetherian coefficient ring \(R\). Goodearl and Letzter give particular emphasis to the use of recently developed techniques from the theory of noncommutative noetherian rings. When \(R\) is an algebra over a field \(k\) on which \(\tau\) and \(\delta\) act trivially, a complete description of the prime ideals of \(S\) is given under the additional assumption that \(\tau ^{-1}\delta \tau = q\delta\) for some nonzero \(q\in k\). This last hypothesis is an abstraction of behavior found in many quantum algebras, including \(q\)-Weyl algebras and coordinate rings of quantum matrices, and specific examples along these lines are considered in detail.

#### Table of Contents

# Table of Contents

## Prime Ideals in Skew and $q$-Skew Polynomial Rings

- Contents v6 free
- 1. Introduction 18 free
- 2. Preliminaries for S = R[y;τ,δ] 613 free
- 3. Tau-delta-prime coefficient rings 1522
- 4. Each prime ideal of S is associated to a unique τ-orbit in specR 2835
- 5. Annihilator primes and induced bimodules 3239
- 6. Prime ideals in quadratic (–1)-skew extensions 4350
- 7. Prime ideals in S associated to infinite orbits. The general case 4653
- 8. Prime ideals in S associated to infinite orbits. The q-skew case 5360
- 9. Prime ideals in S associated to finite orbits. The general case 6067
- 10. Prime ideals in S associated to finite orbits. The q-skew case 6471
- 11. Classification of prime ideals in q-skew extensions 7582
- 12. Irreducible finite dimensional representations of q-skew extensions 8087
- 13. Quantized Weyl algebras 8390
- 14. Prime factors of coordinate rings of quantum matrices 8895
- 15. Chains of prime ideals in iterated Ore extensions 100107
- References 104111