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PRIME AND SEMIPRIME QUANTUM LINEAR SPACE SMASH PRODUCTS

Part of: Radicals and radical properties of rings Rings and algebras arising under various constructions

Published online by Cambridge University Press:  23 July 2020

JASON GADDIS
JASON GADDIS*
Affiliation:
Department of Mathematics, Miami University, 301 S. Patterson Ave., Oxford, OH 45056, USA, e-mail: gaddisj@miamioh.edu
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Abstract

Bosonizations of quantum linear spaces are a large class of pointed Hopf algebras that include the Taft algebras and their generalizations. We give conditions for the smash product of an associative algebra with a bosonization of a quantum linear space to be (semi)prime. These are then used to determine (semi)primeness of certain smash products with quantum affine spaces. This extends Bergen’s work on Taft algebras.

MSC classification

Primary: 16S40: Smash products of general Hopf actions
Secondary: 16N60: Prime and semiprime rings
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 3 , September 2021 , pp. 503 - 514
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Andruskiewitsch, N. and Schneider, H.-J., Lifting of quantum linear spaces and pointed Hopf algebras of order p 3, J. Algebra 209(2) (1998), 658–691.CrossRefGoogle Scholar
Andruskiewitsch, N. and Schneider, H.-J., On the classification of finite-dimensional pointed Hopf algebras, Ann. Math. (2) 171(1) (2010), 375–417.CrossRefGoogle Scholar
Bergen, J., Taft algebras acting on associative algebras, J. Algebra 423 (2015), 422–440.CrossRefGoogle Scholar
Bergen, J. and Grzeszczuk, P., Invariants of U q(sl(2)) and q-skew derivations, J. Pure Appl. Algebra 133(1–2) (1998), 27–38. Ring theory (Miskolc, 1996).CrossRefGoogle Scholar
Bergman, G. M. and Isaacs, I. M., Rings with fixed-point-free group actions, Proc. London Math. Soc. (3) 27 (1973), 69–87.CrossRefGoogle Scholar
Cline, Z., On actions of Drinfel’d doubles on finite dimensional algebras, J. Pure Appl. Algebra 223(8) (2019), 3635–3664.CrossRefGoogle Scholar
Cline, Z. and Gaddis, J., Actions of quantum linear spaces on quantum algebras, J. Algebra 556 (2020), 246–286.CrossRefGoogle Scholar
Cohen, M. and Montgomery, S., Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282(1) (1984), 237–258.CrossRefGoogle Scholar
Cohen, M. and Fishman, D., Hopf algebra actions, J. Algebra 100(2) (1986), 363–379.CrossRefGoogle Scholar
Etingof, P. and Walton, C., Pointed Hopf actions on fields, I, Transform. Groups 20(4) (2015), 985–1013.CrossRefGoogle Scholar
Etingof, P. and Walton, C., Pointed Hopf actions on fields, II, J. Algebra 460 (2016), 253–283.CrossRefGoogle Scholar
Fisher, J. W. and Montgomery, S., Semiprime skew group rings, J. Algebra 52(1) (1978), 241–247.CrossRefGoogle Scholar
Gaddis, J., Won, R. and Yee, D., Discriminants of taft algebra smash products and applications, Algebra Represent. Theory 22(4) (2019), 785–799.CrossRefGoogle Scholar
Goodearl, K. R. and Warfield, R. B., Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 61, 2nd edition (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Kinser, R. and Walton, C., Actions of some pointed Hopf algebras on path algebras of quivers, Algebra Number Theory 10(1) (2016), 117–154.CrossRefGoogle Scholar
Linchenko, V. and Montgomery, S., Semiprime smash products and H-stable prime radicals for PI-algebras, Proc. Amer. Math. Soc. 135(10) (2007), 3091–3098.CrossRefGoogle Scholar
Lomp, C., When is a smash product semiprime? A partial answer, J. Algebra 275(1) (2004), 339–355.CrossRefGoogle Scholar
Lomp, C., On the semiprime smash product question, in Noncommutative rings and their applications, Contemporary Mathematics, vol. 634 (American Mathematical Society, Providence, RI, 2015), 205–222.Google Scholar
Lorenz, M. and Passman, D. S., Two applications of Maschke’s theorem, Comm. Algebra 8(19) (1980), 1853–1866.CrossRefGoogle Scholar
McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings, Graduate Studies in Mathematics, vol. 30, revised edition (American Mathematical Society, Providence, RI, 2001). With the cooperation of L. W. Small.Google Scholar
Passman, D. S., What is a group ring? Amer. Math. Monthly 83(3) (1976), 173–185.CrossRefGoogle Scholar
Passman, D. S., Fixed rings and integrality, J. Algebra 68(2) (1981), 510–519.CrossRefGoogle Scholar
Skryabin, S. and Van Oystaeyen, F., The Goldie Theorem for H-semiprime algebras, J. Algebra 305(1) (2006), 292–320.CrossRefGoogle Scholar
Taft, E. J., The order of the antipode of finite-dimensional Hopf algebra, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 2631–2633.CrossRefGoogle ScholarPubMed
Yanai, T., Automorphic-differential identities and actions of pointed coalgebras on rings, Proc. Amer. Math. Soc. 126(8) (1998), 2221–2228.CrossRefGoogle Scholar