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Primitivity in skew Laurent polynomial rings and related rings. (English) Zbl 0797.16037
Necessary and sufficient conditions for the primitivity of a skew Laurent polynomial ring over a commutative Noetherian ring are given, with explicit constructions of simple faithful modules. In two papers [Glasg. Math. J. 18, 93-97 (1977; Zbl 0347.16020) and ibid. 19, 79-85 (1978; Zbl 0374.16004)], the author gave sufficient conditions for primitivity of formal differential operator rings and skew Laurent polynomial rings, respectively. For formal differential operator rings over commutative Noetherian rings, these conditions were shown to be necessary by K. R. Goodearl and R. B. Warfield jun. [Math. Z. 180, 503-523 (1982; Zbl 0495.16002)]. The results in the present paper are analogous to those obtained by Goodearl and Warfield although the conditions given in the second paper cited above are not necessary.
Necessary and sufficient conditions for primitivity are also given for a related class of rings. A ring in this class is a factor ring of an iterated skew polynomial ring in two indeterminates and, subject to a regularity condition, has two localizations which are skew Laurent polynomial rings. The main examples are the non-Artinian primitive factor rings of both the universal enveloping algebra and the quantum enveloping algebra of the Lie algebra \(\text{sl}(2,\mathbb{C})\).

MSC:
16S36 Ordinary and skew polynomial rings and semigroup rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16S32 Rings of differential operators (associative algebraic aspects)
16S34 Group rings
16W20 Automorphisms and endomorphisms
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