# zbMATH — the first resource for mathematics

Radicals of $$H$$-module algebras. (English. Russian original) Zbl 0776.16013
Algebra Logic 28, No. 6, 462-474 (1989); translation from Algebra Logika 28, No. 6, 705-721 (1989).
Let $$k$$ be a fundamental field. An associative $$k$$-algebra $$H$$, with identity element 1, is said to be a Hopf algebra if on it there are defined algebra homomorphisms $$\Delta: H\to H\otimes H$$, $$\varepsilon: H\to k$$, and a mapping $$s: H\to H$$, satisfying the following conditions: $\text{if }\Delta h= \sum_{(h)} h_{(1)}\otimes h_{(2)}\text{ then}\quad\sum_{(h)}\Delta h_{(1)}\otimes h_{(2)}= \sum_{(h)}h_{(1)}\otimes\Delta h_{(2)},$
$\sum_{(h)} \varepsilon(h_{(1)})h_{(2)}= \sum_{(h)}h_{(1)}\cdot \varepsilon(h_{(2)})=h,\quad \sum_{(h)}s(h_{(1)})h_{(2)}= \sum_{(h)}h_{(1)}s(h_{(2)})= \varepsilon(.)\cdot 1.$ An associative algebra $$A$$ is said to be an $$H$$-module algebra if $$A$$ is a unital right $$H$$-module (where the action of $$H$$ on $$A$$ is written in the form $$a^ h$$) and we have the equality $$(ab)^ h=\sum_{(h)} a^{h_{(1)}} b^{h_{(2)}}$$ for all $$a,b\in A$$, $$h\in H$$. If $$A$$ has an identity, then one imposes additionally the condition $$1^ h=\varepsilon(h)\cdot 1$$ for all $$h\in H$$.
The author studies the theory of classical $$H$$-radicals of associative algebras. The Anderson-Divinsky-Sulinski Theorem (ADS-Theorem) states that if $$A$$ is an associative algebra, $$B\vartriangleleft A$$, and $$\rho$$ is an arbitrary radical, then $$\rho(B)\vartriangleleft A$$. Those $$H$$- radicals for which the ADS-Theorem holds will be called ADS-radicals. It is known that each hypernilpotent $$H$$-radical is an ADS-radical. The author shows that if $$H$$ is a finite dimensional semisimple Hopf algebra, different from the fundamental field $$k$$, then the ADS-Theorem does not hold for $$H$$-radicals. The author then shows that the theory of nil-$$H$$- radicals is basically similar to the classical one, while the Jacobson radical generates an entire spectrum of different $$H$$-radicals. In the final section of the paper, the author defines and studies strongly $$H$$- semisimple algebras.

##### MSC:
 16S90 Torsion theories; radicals on module categories (associative algebraic aspects) 16N20 Jacobson radical, quasimultiplication 16N40 Nil and nilpotent radicals, sets, ideals, associative rings 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16N60 Prime and semiprime associative rings 16S40 Smash products of general Hopf actions
Full Text:
##### References:
 [1] V. A. Andrunakievich and Yu. M. Ryabukhin, Radicals of Algebras and Structural Theory [in Russian], Nauka, Moscow (1989). · Zbl 0663.16017 [2] J. R. Fizher, ”A Jacobson radical for Hopf module algebras,” J. Algebra,34, 217–231 (1975). · Zbl 0306.16012 [3] M. Cohen and S. Montgomery, ”Group-graded rings, smash products, and group actions,” Trans. Am. Math. Soc.,282, No. 1, 237–258 (1984). · Zbl 0533.16001 [4] S. Montgomery, Fixed Rings of Finite Automorphism Groups of Associative Rings, Lecture Notes in Math., No. 818, Springer, Berlin (1980). · Zbl 0449.16001 [5] R. G. Heyneman and M. E. Sweedler, ”Affine Hopf algebras, I.” J. Algebra,13, 192–241 (1969). · Zbl 0203.31601 [6] R. G. Larson and M. E. Sweedler, ”An associative orthogonal bilinear form for Hopf algebras,” Am. J. Math.,91, No. 1, 75–94 (1969). · Zbl 0179.05803 [7] N. V. Loi, ”The A-D-S property for radicals of involution K*-algebras,” Arch. Math. (Basel),49, No. 3, 196–199 (1987). · Zbl 0632.16013 [8] J. Krempa and B. Terlikowska-Oslowska, On graded radical theory. Preprint 11/87, Warsaw (1987). [9] M. Takeuchi, ”Free Hopf algebras generated by coalgebras,” J. Math. Soc. Jpn.,23, No. 4, 561–582 (1971). · Zbl 0217.05902 [10] M. Cohen and D. Fishman, ”Hopf algebra actions,” J. Algebra,100, No. 2, 363–379 (1986). · Zbl 0591.16005 [11] J. Bergen and M. Cohen, ”Actions of commutative Hopf algebras,” Bull. London Math. Soc.,18, 159–164 (1986). · Zbl 0577.16005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.