When is a smash product semiprime? A partial answer

• Type: Article in International Scientific Journal
• Date: 2004

Authors

Lomp, Christian (Author) FCUP

Journal

Title: Journal of Algebra

Vol. 275 No. 1

Pages: 339-355

ISSN: 0021-8693

Publisher: Elsevier

Indexing

ISI Web of Knowledge - 8 Citations

Scopus - 8 Citations

Other information

Authenticus ID: P-000-AJ8

DOI: 10.1016/j.jalgebra.2003.07.017

Abstract (EN): It is in open question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central H-invariant elements of the Martindale ring of quotients of a module algebra form a von Neumann regular and self-injective ring whenever A is semiprime. For a semiprime Goldie P1 H-module, algebra A with central invariants we show that A # H is semiprime if and only if the H-action can be extended to the classical ring of quotients of A if and only if every non-trivial H-stable ideal of A contains a non-zero H-invariant element. In the last section we show that the class of strongly semisimple Hopf algebras is closed under taking Drinfeld twists. Applying some recent results of Etingof and Gelaki we conclude that every semisimple cosemisimple triangular Hopf algebra over a field is strongly semisimple.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 17