Group-graded rings, smash products, and group actions

Let A be a k-algebra graded by a finite group G, with A, 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]*-module algebra. We then study the smash product A#k[G]* it plays a role similar to that played by the skew group ring R * G in the case of group actions, and enables us to obtain results relating the modules over A, A₁, and A#k[G]*. 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 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.