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V. Linchenko, S. Montgomery, L. W. Small, Stable Jacobson Radicals and Semiprime Smash Products, Bulletin of the London Mathematical Society, Volume 37, Issue 6, December 2005, Pages 860–872, https://doi.org/10.1112/S0024609305004662
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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. 2000 Mathematics Subject Classification 16W30, 16N20, 16R99, 16S40.
