Electronic ISBN:  9781470400989 
Product Code:  MEMO/109/521.E 
List Price:  $40.00 
MAA Member Price:  $36.00 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 109; 1994; 106 ppMSC: Primary 16;
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.
ReadershipResearch mathematicians.

Table of Contents

Chapters

1. Introduction

2. Preliminaries for $S$ = $R$[$y; \tau , \delta $]

3. Taudeltaprime coefficient rings

4. Each prime ideal of $S$ is associated to a unique $\tau $orbit in spec$R$

5. Annihilator primes and induced bimodules

6. Prime ideals in quadratic (1)skew extensions

7. Prime ideals in $S$ associated to infinite orbits. The general case

8. Prime ideals in $S$ associated to infinite orbits. The $q$skew case

9. Prime ideals in $S$ associated to finite orbits. The general case

10. Prime ideals in $S$ associated to finite orbits. The $q$skew case

11. Classification of prime ideals in $q$skew extensions

12. Irreducible finite dimensional representations of $q$skew extensions

13. Quantized Weyl algebras

14. Prime factors of coordinate rings of quantum matrices

15. Chains of prime ideals in iterated Ore extensions


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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.
Research mathematicians.

Chapters

1. Introduction

2. Preliminaries for $S$ = $R$[$y; \tau , \delta $]

3. Taudeltaprime coefficient rings

4. Each prime ideal of $S$ is associated to a unique $\tau $orbit in spec$R$

5. Annihilator primes and induced bimodules

6. Prime ideals in quadratic (1)skew extensions

7. Prime ideals in $S$ associated to infinite orbits. The general case

8. Prime ideals in $S$ associated to infinite orbits. The $q$skew case

9. Prime ideals in $S$ associated to finite orbits. The general case

10. Prime ideals in $S$ associated to finite orbits. The $q$skew case

11. Classification of prime ideals in $q$skew extensions

12. Irreducible finite dimensional representations of $q$skew extensions

13. Quantized Weyl algebras

14. Prime factors of coordinate rings of quantum matrices

15. Chains of prime ideals in iterated Ore extensions