Vol. 2, No. 4, 2008

Download this article
Download this article For screen
For printing
Recent Issues

Volume 18
Issue 6, 1039–1219
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author index
To appear
Other MSP journals
Group actions and rational ideals

Martin Lorenz

Vol. 2 (2008), No. 4, 467–499

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.

algebraic group, rational action, prime ideal, rational ideal, primitive ideal, generic ideal, extended centroid, Amitsur–Martindale ring of quotient
Mathematical Subject Classification 2000
Primary: 16W22
Secondary: 16W35, 17B35
Received: 24 January 2008
Accepted: 28 April 2008
Published: 15 June 2008
Martin Lorenz
Department of Mathematics
Temple University
Philadelphia, PA 19122-6094
United States