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  • Cited by 8
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Eryashkin, M. S. 2016. Invariants of the action of a semisimple Hopf algebra on PI-algebra. Russian Mathematics, Vol. 60, Issue. 8, p. 17.


    Gordienko, A. S. 2016. Co-stability of Radicals and Its Applications to PI-Theory. Algebra Colloquium, Vol. 23, Issue. 03, p. 481.


    Karasik, Yaakov 2016. Kemer's theory for H-module algebras with application to the PI exponent. Journal of Algebra, Vol. 457, p. 194.


    Skryabin, Serge 2015. Invariant subrings and Jacobson radicals of Noetherian Hopf module algebras. Israel Journal of Mathematics, Vol. 207, Issue. 2, p. 881.


    Gordienko, A.S. 2013. Amitsur’s conjecture for associative algebras with a generalized Hopf action. Journal of Pure and Applied Algebra, Vol. 217, Issue. 8, p. 1395.


    Skryabin, Serge 2011. Structure of H-Semiprime Artinian Algebras. Algebras and Representation Theory, Vol. 14, Issue. 5, p. 803.


    Linchenko, V. 2007. Relations Between Algebras and Their Subalgebras of Invariants. Communications in Algebra, Vol. 35, Issue. 6, p. 1834.


    Skryabin, Serge and Van Oystaeyen, Freddy 2006. The Goldie Theorem for H-semiprime algebras. Journal of Algebra, Vol. 305, Issue. 1, p. 292.


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  • Bulletin of the London Mathematical Society, Volume 37, Issue 6
  • December 2005, pp. 860-872

STABLE JACOBSON RADICALS AND SEMIPRIME SMASH PRODUCTS

  • V. LINCHENKO (a1), S. MONTGOMERY (a2) and L. W. SMALL (a3)
  • DOI: http://dx.doi.org/10.1112/S0024609305004662
  • Published online: 12 December 2005
Abstract

We prove that if H is a finite-dimensional semisimple Hopf algebra acting on a PI-algebra R of characteristic 0, and R is either affine or algebraic over k, then the Jacobson radical of R is H-stable. Under the same hypotheses, we show that the smash product algebra R#H is semiprimitive provided that R is H-semiprime. More generally we show that the ‘finite’ Jacobson radical is H-stable, and that R#H is semiprimitive provided that R is H-semiprimitive and all irreducible representations of R are finite-dimensional. We also consider R#H when R is an FCR-algebra. Finally, we prove a general relationship between stability of the radical and semiprimeness of R#H; in particular if for a given H, any action of H stabilizes the Jacobson radical, then also any action of H stabilizes the prime radical.

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Bulletin of the London Mathematical Society
  • ISSN: 0024-6093
  • EISSN: 1469-2120
  • URL: /core/journals/bulletin-of-the-london-mathematical-society
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