Abstract
We construct the H-von Neumann regular radical for H-module algebras and show that it is an H-radical property. We obtain that the Jacobson radical of a twisted graded algebra is a graded ideal. For a twisted H-module algebra R, we also show that r j (R# σ H) = r Hj (R)# σ H and the Jacobson radical of R is stable, when k is an algebraically closed field or there exists an algebraic closure F of k such that r j (R⊗F) = r j (R) ⊗ F, where H is a finite-dimensional, semisimple, cosemisimple, commutative or cocommutative Hopf algebra over k. In particular, we answer two questions of J. R. Fisher.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Aslam and A. M. Zaidi, The matrix equation in radicals, Studia Sci. Math. Hungar., 28 (1993), 447–452.
M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. AMS, 282 (1984), 237–258.
J. R. Fisher, The Jacobson radicals for Hopf module algebras, J. Algebra, 25 (1975), 217–221.
J. Krempa, Logical connections among some open problems in noncommutative rings, Fund. Math., 76 (1972), 121–130.
W. G. Leavitt, A note on matric-extensibility and ADS condition, Studia Sci. Math. Hungar., 32 (1996), 407–414.
W. G. Leavitt, Matric-extensible radicals, in Proceedings of Int. Conference on Rings and Radicals, eds. B. J. Gardner, S. X. Liu and R. Wiegandt (Shijiazhuang, 1994).
S. Montgomery and H. J. Schneider, Hopf crossed products, rings of quotients, and prime ideals, Advances in Mathematics, 112 (1995), 1–55.
S. Montgomery, Hopf Algebras and their Actions on Rings, CBMS Number 82, Published by AMS (1992).
R. E. Propes, The radical equation P(A n) = (P(A))n, Proc. Edinburgh Math. Soc., 19 (1974), 257–259.
C. Nastasescu and F. van Oystaeyen, Graded Ring Theory, North-Holland Publishing Company (1982).
D. S. Passman, The Algebraic Structure of Group Rings, John Wiley and Sons (New York, 1977).
D. E. Radford, The trace function and Hopf algebras, J. Algebra, 163 (1994), 583–622.
A. D. Sands, Radicals and Morita contexts, J. Algebra, 24 (1973), 335–345.
M. E. Sweedler, Hopf Algebras, Benjamin (New York, 1969).
F. A. Szász, Radicals of Rings, John Wiley and Sons (New York, 1982).
Shouchuan Zhang, The radicals of Hopf module algebras, Chinese Ann. Mathematics, Ser B, 18 (1997), 495–502.
Shouchuan Zhang, The Baer and Jacobson radicals of crossed products, Acta Math. Hungar., 78 (1998), 11–24.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zhang, S. The Von Neumann Regular Radical and Jacobson Radical of Crossed Products. Acta Mathematica Hungarica 86, 319–333 (2000). https://doi.org/10.1023/A:1006723709818
Issue Date:
DOI: https://doi.org/10.1023/A:1006723709818