Group-graded rings, smash products, and group actions

Cohen, M.; Montgomery, S.

doi:10.1090/S0002-9947-1984-0728711-4

Group-graded rings, smash products, and group actions
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by M. Cohen and S. Montgomery PDF

Trans. Amer. Math. Soc. 282 (1984), 237-258 Request permission

Addendum: Trans. Amer. Math. Soc. 300 (1987), 810-811.

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 $A\# k{[G]^ \ast }$; 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 $A\# k{[G]^ \ast }$. 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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