Algebra & Number Theory

Group actions and rational ideals

Martin Lorenz

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We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur–Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting.

Our main result concerns rational actions of an affine algebraic group G on R. Working over an algebraically closed base field, we prove an existence and uniqueness result for generic rational ideals in the sense of Dixmier: for every G-rational ideal I of R, the closed subset of the rational spectrum RatR that is defined by I is the closure of a unique G-orbit in RatR. Under additional Goldie hypotheses, this was established earlier by Mœglin and Rentschler (in characteristic 0) and by Vonessen (in arbitrary characteristic), answering a question of Dixmier.

Article information

Algebra Number Theory, Volume 2, Number 4 (2008), 467-499.

Received: 24 January 2008
Accepted: 28 April 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16W22: Actions of groups and semigroups; invariant theory
Secondary: 16W35 17B35: Universal enveloping (super)algebras [See also 16S30]

algebraic group rational action prime ideal rational ideal primitive ideal generic ideal extended centroid Amitsur–Martindale ring of quotient


Lorenz, Martin. Group actions and rational ideals. Algebra Number Theory 2 (2008), no. 4, 467--499. doi:10.2140/ant.2008.2.467.

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