Rings graded by polycyclic-by-finite groups
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- by William Chin and Declan Quinn
- Proc. Amer. Math. Soc. 102 (1988), 235-241
- DOI: https://doi.org/10.1090/S0002-9939-1988-0920979-3
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Abstract:
We use the duality between group gradings and group actions to study polycyclic-by-finite group-graded rings. We show that, for such rings, graded Noetherian implies Noetherian and relate the graded Krull dimension to the Krull dimension. In addition we find a bound on the length of chains of prime ideals not containing homogeneous elements when the grading group is nilpotent-by-finite. These results have suitable corollaries for strongly group-graded rings. Our work extends several results on skew group rings, crossed products and group-graded rings.References
- Allen D. Bell, Localization and ideal theory in Noetherian strongly group-graded rings, J. Algebra 105 (1987), no. 1, 76–115. MR 871748, DOI 10.1016/0021-8693(87)90181-5 M. Van den Bergh, A duality theorem for Hopf module algebras, Methods in Ring Theory, NATO ASI Series, vol. 129, Reidel, Dordrecht, 1983. G. Bergman, Homogeneous elements and prime ideals in ${\mathbf {Z}}$-graded rings, preprint.
- Robert J. Blattner and Susan Montgomery, A duality theorem for Hopf module algebras, J. Algebra 95 (1985), no. 1, 153–172. MR 797661, DOI 10.1016/0021-8693(85)90099-7
- William Chin, Prime ideals in differential operator rings and crossed products of infinite groups, J. Algebra 106 (1987), no. 1, 78–104. MR 878469, DOI 10.1016/0021-8693(87)90022-6
- M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), no. 1, 237–258. MR 728711, DOI 10.1090/S0002-9947-1984-0728711-4
- Robert Gordon and J. C. Robson, Krull dimension, Memoirs of the American Mathematical Society, No. 133, American Mathematical Society, Providence, R.I., 1973. MR 0352177
- Martin Lorenz and D. S. Passman, Prime ideals in crossed products of finite groups, Israel J. Math. 33 (1979), no. 2, 89–132. MR 571248, DOI 10.1007/BF02760553
- Constantin Năstăsescu, Group rings of graded rings. Applications, J. Pure Appl. Algebra 33 (1984), no. 3, 313–335. MR 761636, DOI 10.1016/0022-4049(84)90065-3
- James Osterburg, Smash products and $G$-Galois actions, Proc. Amer. Math. Soc. 98 (1986), no. 2, 217–221. MR 854022, DOI 10.1090/S0002-9939-1986-0854022-X
- Declan Quinn, Group-graded rings and duality, Trans. Amer. Math. Soc. 292 (1985), no. 1, 155–167. MR 805958, DOI 10.1090/S0002-9947-1985-0805958-0
- P. F. Smith, Corrigendum: On the dimension of group rings (Proc. London Math. Soc. (3) 25 (1972), 288–302), Proc. London Math. Soc. (3) 27 (1973), 766–768. MR 325671, DOI 10.1112/plms/s3-27.4.766-s
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Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 235-241
- MSC: Primary 16A03,; Secondary 16A27,16A33,16A55
- DOI: https://doi.org/10.1090/S0002-9939-1988-0920979-3
- MathSciNet review: 920979