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Prime Ideals in Skew and $$q$$-Skew Polynomial Rings
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Memoirs of the American Mathematical Society
1994; 106 pp; softcover
Volume: 109
ISBN-10: 0-8218-2583-6
ISBN-13: 978-0-8218-2583-9
List Price: US$38 Individual Members: US$22.80
Institutional Members: US\$30.40
Order Code: MEMO/109/521

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.

• Introduction
• Preliminaries for $$S=R[y;\tau ,\delta ]$$
• Tau-delta-prime coefficient rings
• Each prime ideal of $$S$$ is associated to a unique $$\tau$$-orbit in $$\operatorname{spec}R$$
• Annihilator primes and induced bimodules
• Prime ideals in quadratic $$(-1)$$-skew extensions
• Prime ideals in $$S$$ associated to infinite orbits. The general case
• Prime ideals in $$S$$ associated to infinite orbits. The $$q$$-skew case
• Prime ideals in $$S$$ associated to finite orbits. The general case
• Prime ideals in $$S$$ associated to finite orbits. The $$q$$-skew case
• Classification of prime ideals in $$q$$-skew extensions
• Irreducible finite dimensional representations of $$q$$-skew extensions
• Quantized Weyl algebras
• Prime factors of coordinate rings of quantum matrices
• Chains of prime ideals in iterated Ore extensions
• References