Geodesic
Radicals around Köthe's problem
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2004), pp. 76-84 
S. Tumurbat; R. Wiegandt. Radicals around Köthe's problem. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2004), pp. 76-84. http://geodesic.mathdoc.fr/item/BASM_2004_1_a8/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Radicals $\gamma$ will be studied for which the condition "$A[x] \in\gamma$ for all nil rings $A$" is equivalent to the positive solution of Köthe's Problem ($A[x]$ is Jacobson radical for all nil rings $A$, in Krempa's formulation). The closer $\gamma$ is to the Jacobson radical, the better approximation of the positive solution is obtained. Seeking, however, for a negative solution, possibly large radicals $\gamma$ are of interest. In this note such large radicals will be studied.

[1] Beidar K. I., Fong Y., Puczyłowski E. R., “A polynomial ring over a nil ring cannot be homomorphically mapped onto a ring with nonzero idempotents”, J. Algebra, 238 (2001), 389–399 | DOI | MR | Zbl

[2] Beidar K. I., Puczyłowski E. R., Wiegandt R., “Radicals and polynomial rings”, J. Austral. Math. Soc., 71 (2002), 1–7 | MR

[3] Drazin M. P., Roberts M. L., “Polynomial, multiplicative and special radicals”, Comm. in Algebra, 28 (2000), 3073–3093 | DOI | MR | Zbl

[4] Gardner B. J., Wiegandt R., Radical theory of rings, Marcel Dekker, 2004 | MR

[5] Krempa J., “Logical connections among some open problems in non-commutative rings”, Fund. Math., 76 (1972), 121–130 | MR

[6] McConnell N. R., Stokes T., “Rings having simple adjoint semigroup”, Contemporary Math., 273, 2001, 203–208 | MR | Zbl

[7] Sakhajev I. I., “Negative solution hypothesis of Koethe and some problems connected with it”, Abstract of Talks. Algebra Conference Venezia, 2002, 37–38

[8] Sands A. D., “Radicals and Morita contexts”, J. Algebra, 24 (1973), 335–345 | DOI | MR | Zbl

[9] Tumurbat S., “The radicalness of polynomial rings over nil rings”, Math. Pannonica, 13 (2002), 191–199 | MR | Zbl