×

zbMATH — the first resource for mathematics

Group-graded rings, smash products, and group actions. (English) Zbl 0533.16001
Let \(k\) be a commutative ring with unit and \(G\) be a finite group (not necessarily abelian). In this paper, the authors consider \(G\)-graded \(k\)-algebras as \(k[G]^*\)-module algebras where \(k[G]^*\) is the Hopf algebra dual to the usual group ring \(k[G]\). This point of view yields results for graded algebras analogous to those for algebras on which \(G\) acts as a group of automorphisms. Most important is the duality theorem for coactions which states that if \(A\) is \(G\)-graded, then there is a \(G\)-action on \(A\#k[G]^*\), and \((A\#k[G]^*)^*G\cong M_n(A)\), the \(n\times n\) matrix ring over \(A\). The authors then apply this theorem to problems about graded rings such as determining the relationship of the graded Jacobson (prime) radical to the usual Jacobson (prime) radical, comparing the prime ideals of \(A\) and \(A\#k[G]^*\), and showing incomparability for primes in \(A\) and \(A_1\). The paper includes all definitions and propositions needed for the new material, often with proofs.

MSC:
16W50 Graded rings and modules (associative rings and algebras)
16T05 Hopf algebras and their applications
16W20 Automorphisms and endomorphisms
16S34 Group rings
16Dxx Modules, bimodules and ideals in associative algebras
46L40 Automorphisms of selfadjoint operator algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. A. Amitsur, Rings of quotients and Morita contexts, J. Algebra 17 (1971), 273 – 298. · Zbl 0221.16014
[2] G. Bergman, Groups acting on rings, group graded rings, and beyond (preprint).
[3] -, On Jacobson radicals of graded rings (preprint).
[4] S. U. Chase, D. K. Harrison, and Alex Rosenberg, Galois theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc. No. 52 (1965), 15 – 33. · Zbl 0143.05902
[5] Miriam Cohen, A Morita context related to finite automorphism groups of rings, Pacific J. Math. 98 (1982), no. 1, 37 – 54. · Zbl 0488.16024
[6] Miriam Cohen and Louis H. Rowen, Group graded rings, Comm. Algebra 11 (1983), no. 11, 1253 – 1270. · Zbl 0522.16001
[7] Everett C. Dade, Group-graded rings and modules, Math. Z. 174 (1980), no. 3, 241 – 262. · Zbl 0424.16001
[8] Carl Faith, Algebra: rings, modules and categories. I, Springer-Verlag, New York-Heidelberg, 1973. Die Grundlehren der mathematischen Wissenschaften, Band 190. · Zbl 0266.16001
[9] J. M. G. Fell, Induced representations and Banach *-algebraic bundles, Lecture Notes in Mathematics, Vol. 582, Springer-Verlag, Berlin-New York, 1977. With an appendix due to A. Douady and L. Dal Soglio-Hérault. · Zbl 0372.22001
[10] John R. Fisher, A Jacobson radical for Hopf module algebras, J. Algebra 34 (1975), 217 – 231. · Zbl 0306.16012
[11] Joe W. Fisher and Susan Montgomery, Semiprime skew group rings, J. Algebra 52 (1978), no. 1, 241 – 247. · Zbl 0373.16004
[12] I. N. Herstein, Noncommutative rings, The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. · Zbl 0177.05801
[13] A. Joseph and L. W. Small, An additivity principle for Goldie rank, Israel J. Math. 31 (1978), no. 2, 105 – 114. · Zbl 0395.17010
[14] Magnus B. Landstad, Duality for dual covariance algebras, Comm. Math. Phys. 52 (1977), no. 2, 191 – 202. · Zbl 0362.46046
[15] Martin Lorenz and D. S. Passman, Prime ideals in crossed products of finite groups, Israel J. Math. 33 (1979), no. 2, 89 – 132. , https://doi.org/10.1007/BF02760553 Martin Lorenz and D. S. Passman, Addendum: ”Prime ideals in crossed products of finite groups”, Israel J. Math. 35 (1980), no. 4, 311 – 322. · Zbl 0439.16010
[16] Martin Lorenz, Susan Montgomery, and L. W. Small, Prime ideals in fixed rings. II, Comm. Algebra 10 (1982), no. 5, 449 – 455. · Zbl 0482.16027
[17] Susan Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. · Zbl 0449.16001
[18] Susan Montgomery, Prime ideals in fixed rings, Comm. Algebra 9 (1981), no. 4, 423 – 449. · Zbl 0453.16019
[19] Yoshiomi Nakagami, Dual action on a von Neumann algebra and Takesaki’s duality for a locally compact group, Publ. Res. Inst. Math. Sci. 12 (1976/77), no. 3, 727 – 775. · Zbl 0363.46062
[20] Yoshiomi Nakagami and Masamichi Takesaki, Duality for crossed products of von Neumann algebras, Lecture Notes in Mathematics, vol. 731, Springer, Berlin, 1979. · Zbl 0423.46051
[21] Constantin Năstăsescu, Strongly graded rings of finite groups, Comm. Algebra 11 (1983), no. 10, 1033 – 1071. · Zbl 0522.16002
[22] W. K. Nicholson and J. F. Watters, Normal radicals and normal classes of rings, J. Algebra 59 (1979), no. 1, 5 – 15. · Zbl 0413.16005
[23] Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. · Zbl 0368.16003
[24] D. S. Passman, Fixed rings and integrality, J. Algebra 68 (1981), no. 2, 510 – 519. , https://doi.org/10.1016/0021-8693(81)90277-5 D. S. Passman, Erratum: ”Fixed rings and integrality”, J. Algebra 73 (1981), no. 1, 273. · Zbl 0484.16022
[25] -, It’s essentially Maschke’s theorem, Rocky Mountain J. Math. 13 (1978), 37-54. · Zbl 0525.16022
[26] Gert K. Pedersen, \?*-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. · Zbl 0416.46043
[27] Şerban Strătilă, Dan Voiculescu, and Laszló Zsidó, On crossed products. I, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 10, 1411 – 1449. · Zbl 0367.46060
[28] Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. · Zbl 0194.32901
[29] Masamichi Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249 – 310. · Zbl 0268.46058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.