A Morita context related to finite automorphism groups of rings

Cohen, M.

doi:10.2140/pjm.1982.98.37

Cited by 31 publications

(12 citation statements)

References 17 publications

(23 reference statements)

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“…For a finite group G acting on a commutative ring A, Chase, Harrison, and Rosenberg [4] showed that there is a Morita context [A G , V, W, A * G] associated with the fixed ring A G and the skew group ring A * G, using V = AcAA • G and W = A * GAA". This context was studied for noncommutative rings by M. Cohen in [5]. We show in this section that an analogous situation holds for coactions: if A is graded by G, there is a Morita context associated to A#k[G]* and AI.…”

Section: Modules Morita Contexts and Gradingsmentioning

confidence: 82%

Group-graded rings, smash products, and group actions

Cohen¹,

Montgomery²

1984

Trans. Amer. Math. Soc.

266

112

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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.

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Section: Modules Morita Contexts and Gradingsmentioning

confidence: 82%

Group-graded rings, smash products, and group actions

Cohen¹,

Montgomery²

1984

Trans. Amer. Math. Soc.

266

112

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“…Similarly Pi n T= (0) as Pi is a minimal prime; thus (Plh, ... ,(Pk)T are precisely the primes of R, and rk(AI/pJ = rk«R/pJT)' Thus the theorem follows by Proposition 7. 5. 0 We do not know whether it would suffice to assume that A / P is Goldie, rather than A/Pc is Goldie.…”

Section: Proof For Any Graded Primementioning

confidence: 99%

Group-Graded Rings, Smash Products, and Group Actions

Cohen¹,

Montgomery²

1984

Transactions of the American Mathematical Society

111

165

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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]*-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.

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“…[12,13] Let R be a semiprime ring and G a group of ring automorphisms of R. If is semiprime, then is semiprime.…”

Section: Preliminarymentioning

confidence: 99%