Group-graded rings, smash products, and group actions

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]*-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, A I' 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 A I' 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.

Crossed Products and Inner Actions of Hopf Algebras

“…Exactly the same proof given in [4] works here. D Note that the G-grading of B #CT(fcG) is exactly that arising from the right fcG-comodule structure on B #CT (fcG) discussed above.…”

“…Now if the characteristic of fc divides the order of G, fcG is not completely reducible as a left G-module. G Despite this example, it is always true that A semisimple implies A # (fcG)* is semisimple, using other methods [4]. We are able to show that A semiprime implies A #" H semiprime in the two cases of Corollary 6.6.…”

“…D Note that the G-grading of B #CT(fcG) is exactly that arising from the right fcG-comodule structure on B #CT (fcG) discussed above. Also note that the proof of duality for coactions of finite groups in [4] does not carry over to the twisted case. EXAMPLE 5.7.…”

On classical Krull dimension of group-graded rings

“…Denote by cl-K-dim(-R) the classical Krull dimension of R. For any ordinal a and positive integer n, we introduce ordinals a n , setting c*i = a + 1, a n+i = (a + l)(a n + 1)-We shall use the results on prime ideals due to Cohen and Montgomery [3] and prove the following theorem. …”

“…< a. Then there exists the least ordinal a such that S a = S. This ordinal is called the classical Krull dimension of R. If it is finite, then it is also equal to the right Krull dimension of R defined on the lattice of right ideals of R (see [5, Chapter 6]).Denote by cl-K-dim(-R) the classical Krull dimension of R. For any ordinal a and positive integer n, we introduce ordinals a n , setting c*i = a + 1, a n+i = (a + l)(a n + 1)-We shall use the results on prime ideals due to Cohen and Montgomery [3] and prove the following theorem. …”