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Group-graded rings, smash products, and group actions
- Source :
- Transactions of the American Mathematical Society. 282:237-258
- Publication Year :
- 1984
- Publisher :
- American Mathematical Society (AMS), 1984.
-
Abstract
- Let A A be a k k -algebra graded by a finite group G G , with A 1 {A_1} the component for the identity element of G G . We consider such a grading as a “coaction” by G G , in that A A is a k [ G ] ∗ k{[G]^ \ast } -module algebra. We then study the smash product A # k [ G ] ∗ A\# k{[G]^ \ast } ; it plays a role similar to that played by the skew group ring R ∗ G R\, \ast \,G in the case of group actions, and enables us to obtain results relating the modules over A , A 1 A,\,{A_1} , and A # k [ G ] ∗ A\# k{[G]^ \ast } . After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of A A and A 1 {A_1} . In particular we generalize Lorenz and Passman’s theorem on incomparability of primes in crossed products. We also answer a question of Bergman on graded Jacobson radicals.
Details
- ISSN :
- 10886850 and 00029947
- Volume :
- 282
- Database :
- OpenAIRE
- Journal :
- Transactions of the American Mathematical Society
- Accession number :
- edsair.doi...........642f23f568cfd4b635b8940dd75ab3ff