Abstract
Let H be a Hopf algebra over a field. It is proved that every H-semiprime right artinian left H-module algebra A is quasi-Frobenius and H-semisimple. If H grows slower than exponentially, then all H-equivariant A-modules are shown to be A-projective. With the additional assumption that H is cosemisimple it is proved that the Jacobson radical of any right artinian left H-module algebra is stable under the action of H.
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References
Bahturin, Yu.A., Linchenko, V.: Identities of algebras with actions of Hopf algebras. J. Algebra 202, 634–654 (1998)
Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans. Am. Math. Soc. 95, 466–488 (1960)
Beck, I.: Projective and free modules. Math. Z. 129, 231–234 (1972)
Block, R.E.: Determination of the differentiably simple rings with a minimal ideal. Ann. Math. 90, 433–459 (1969)
Cohen, M., Fischman, D.: Hopf algebra actions. J. Algebra 100, 363–379 (1986)
Doi, Y.: On the structure of relative Hopf modules. Commun. Algebra 11, 243–255 (1983)
Donkin, S.: The restriction of the regular module for a quantum group. In: Algebraic Groups and Lie Groups. A Volume of Papers in Honor of the late R.W. Richardson. Aust. Math. Soc. Lect. Ser., vol. 9, pp. 183–188. Cambridge University Press, Cambridge, UK (1997)
Eilenberg, S., Nakayama, T.: On the dimension of modules and algebras. II (Frobenius algebras and quasi-Frobenius rings.). Nagoya Math. J. 9, 1–16 (1955)
Kasch, F.: Moduln und Ringe. Teubner, Leipzig (1977)
Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension, Graduate Studies in Mathematics, vol. 22. American Mathematical Society, Providence, RI (2000)
Larson, R.G., Sweedler, M.E.: An associative orthogonal bilinear form for Hopf algebras. Am. J. Math. 91, 75–94 (1969)
Linchenko, V.: Nilpotent subsets of Hopf module algebras. In: Groups, Rings, Lie and Hopf Algebras, Proceedings of the 2001 St. John’s Conference, pp. 121–127. Kluwer, Norwell, MA (2003)
Linchenko, V., Montgomery, S., Small, L.W.: Stable Jacobson radicals and semiprime smash products. Bull. Lond. Math. Soc. 37, 860–872 (2005)
Linchenko, V., Montgomery, S.: Semiprime smash products and H-stable prime radicals for PI-algebras. Proc. Am. Math. Soc. 135, 3091–3098 (2007)
Masuoka, A.: Freeness of Hopf algebras over coideal subalgebras. Commun. Algebra 20, 1353–1373 (1992)
Montgomery, S.: Hopf algebras and their actions on rings. In: CBMS Regional Conference Series in Mathematics, vol. 82. American Mathematical Society, Providence, RI (1993)
Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8, 177–206 (2003)
Příhoda, P.: Projective modules are determined by their radical factors. J. Pure Appl. Algebra 210, 827–835 (2007)
Radford, D.E.: Freeness (projectivity) criteria for Hopf algebras over Hopf subalgebras. J. Pure Appl. Algebra 11, 15–28 (1977)
Skryabin, S.: Projectivity and freeness over comodule algebras. Trans. Am. Math. Soc. 359, 2597–2623 (2007)
Skryabin, S.: Hopf algebra orbits on the prime spectrum of a module algebra. Algebr. Represent. Theory 13, 1–31 (2010)
Skryabin, S., Van Oystaeyen, F.: The Goldie theorem for H-semiprime algebras. J. Algebra 305, 292–320 (2006)
Strade, H.: The Classification of the Simple Lie Algebras over Fields with Positive Characteristic. De Gruyter, Berlin (2004)
Sweedler, M.E.: Hopf Algebras. Benjamin, New York (1969)
Takeuchi, M.: Relative Hopf modules—equivalences and freeness criteria. J. Algebra 60, 452–471 (1979)
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This work was supported by the RFBR grant 10-01-00431.
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Skryabin, S. Structure of H-Semiprime Artinian Algebras. Algebr Represent Theor 14, 803–822 (2011). https://doi.org/10.1007/s10468-010-9216-8
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DOI: https://doi.org/10.1007/s10468-010-9216-8