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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Group-graded rings, smash products, and group actions


Authors: M. Cohen and S. Montgomery
Journal: Trans. Amer. Math. Soc. 282 (1984), 237-258
MSC: Primary 16A03; Secondary 16A12, 16A24, 16A66, 16A72, 46L99
Addendum: Trans. Amer. Math. Soc. 300 (1987), 810-811.
MathSciNet review: 728711
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Abstract: Let $ A$ be a $ k$-algebra graded by a finite group $ G$, with $ {A_1}$ the component for the identity element of $ G$. We consider such a grading as a ``coaction'' by $ G$, in that $ A$ is a $ k{[G]^ \ast }$-module algebra. We then study the smash product $ A\char93 k{[G]^ \ast }$; it plays a role similar to that played by the skew group ring $ R\, \ast \,G$ in the case of group actions, and enables us to obtain results relating the modules over $ A,\,{A_1}$, and $ A\char93 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$ and $ {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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0728711-4
PII: S 0002-9947(1984)0728711-4
Article copyright: © Copyright 1984 American Mathematical Society