Groupgraded rings, smash products, and group actions
Authors:
M. Cohen and S. Montgomery
Journal:
Trans. Amer. Math. Soc. 282 (1984), 237258
MSC:
Primary 16A03; Secondary 16A12, 16A24, 16A66, 16A72, 46L99
Addendum:
Trans. Amer. Math. Soc. 300 (1987), 810811.
MathSciNet review:
728711
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Abstract: Let be a algebra graded by a finite group , with the component for the identity element of . We consider such a grading as a ``coaction'' by , in that is a module algebra. We then study the smash product ; it plays a role similar to that played by the skew group ring in the case of group actions, and enables us to obtain results relating the modules over , and . 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 and . 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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 [1]
 S. A. Amitsur, Rings of quotients and Morita contexts, J. Algebra 17 (1971), 273298. MR 0414604 (54:2704)
 [2]
 G. Bergman, Groups acting on rings, group graded rings, and beyond (preprint).
 [3]
 , On Jacobson radicals of graded rings (preprint).
 [4]
 S. Chase, D. Harrison and A. Rosenberg, Galois theory and cohomology of commutative rings, Mem. Amer. Math. Soc., No. 52, 1965. MR 0195922 (33:4118)
 [5]
 M. Cohen, A Morita context related to finite automorphism groups of rings, Pacific J. Math. 98 (1982), 3754. MR 644936 (83g:16063)
 [6]
 M. Cohen and L. Rowen, Group graded rings, Comm. Algebra 11 (1983), 12531270. MR 696990 (85b:16002)
 [7]
 E. C. Dade, Group graded rings and modules, Math. Z. 174 (1980), 241262. MR 593823 (82c:16028)
 [8]
 C. Faith, Algebra: rings, modules, and categories. I, SpringerVerlag, New York, 1973. MR 0366960 (51:3206)
 [9]
 J. M. G. Fell, Induced representations and Banach *algebraic bundles, Lecture Notes in Math., vol. 582, SpringerVerlag, Berlin and New York, 1977. MR 0457620 (56:15825)
 [10]
 J. R. Fisher, A Jacobson radical for Hopf module algebras, J. Algebra 34 (1975), 217231. MR 0366963 (51:3209)
 [11]
 J. W. Fisher and S. Montgomery, Semiprime skew group rings, J. Algebra 52 (1978), 241247. MR 0480616 (58:772)
 [12]
 I. N. Herstein, Noncommutative rings, Carus Math. Monographs, No. 15, Math. Assoc. Amer., 1968. MR 0227205 (37:2790)
 [13]
 A. Joseph and L. W. Small, An additivity principle for Goldie rank, Israel J. Math. 31 (1978), 105114. MR 516246 (80j:17005)
 [14]
 M. B. Landstad, Duality for dual covariance algebras, Comm. Math. Phys. 52 (1977), 191202. MR 0450456 (56:8750)
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 M. Lorenz and D. S. Passman, Prime ideals in crossed products of finite groups, Israel J. Math. 33 (1979), 89132. MR 571248 (82k:16042a)
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 M. Lorenz, S. Montgomery and L. W. Small, Prime ideals in fixed rings. II, Comm. Algebra 10 (1982), 449455. MR 647831 (83b:16031)
 [17]
 S. Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Math., vol. 818, SpringerVerlag, Berlin and New York, 1980. MR 590245 (81j:16041)
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 , Prime ideals in fixed rings, Comm. Algebra 9 (1981), 423449. MR 605031 (82c:16034)
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 Y. Nakagami, Dual action on a von Neumann algebra and Takesaki's duality for a locally compact group, Publ. Res. Inst. Math. Sci. 12 (1977), 727775. MR 0458190 (56:16393)
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 Y. Nakagami and M. Takesaki, Duality for crossed products of von Neumann algebras, Lecture Notes in Math., vol. 731, SpringerVerlag, Berlin and New York, 1979. MR 546058 (81e:46053)
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 C. Nastacescu, Strongly graded rings of finite groups, Comm. Algebra 11 (1983). MR 700723 (84k:16004)
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 W. K. Nicholson and J. F. Waters, Normal radicals and normal classes of rings, J. Algebra 59 (1979), 515. MR 541666 (80h:16007)
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 [25]
 , It's essentially Maschke's theorem, Rocky Mountain J. Math. 13 (1978), 3754.
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 M. Sweedler, Hopf algebras, Benjamin, New York, 1969. MR 0252485 (40:5705)
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 M. Takesaki, Duality in crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249310. MR 0438149 (55:11068)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198407287114
PII:
S 00029947(1984)07287114
Article copyright:
© Copyright 1984
American Mathematical Society
