A Note on Polynomial Rings over Nil Rings
Let R be a nil ring with p R = 0 for some prime number p. We show that the polynomial ring R[x,y] in two commuting indeterminates x, y over R cannot be homomorphically mapped onto a ring with identity. This extends, in finite characteristic case, a result obtained by Smoktunowicz  and gives a new approximation, in that case, of a positive solution of Köthe’s problem.
KeywordsNil ring polynomial ring Brown-McCoy radical Köthe’s problem
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- Ferrero, M. Unitary strongly prime rings and ideals. Proceedings of the 35th Symposium on Ring Theory and Representation Theory (Okayama, 2002), 101–111, Symp. Ring Theory Representation Theory Organ. Comm., Okayama, 2003.Google Scholar
- Puczyłowski, E.R. Some questions concerning radicals of associative rings. Theory of radicals (Szekszard, 1991), 209–227, Colloq. Math. Soc. Janos Bolyai, 61, North-Holland, Amsterdam, 1993.Google Scholar
- Puczyłowski, E.R. Some results and questions on nil rings. 15th School of Algebra (Portuguese) (Canela, 1998). Mat. Contemp. 16 (1999), 265–280.Google Scholar
- Smoktunowicz, A. R[x,y] is Brown-McCoy radical if R[x] is Jacobson radical. Proc. 3rd Int. Algebra Conf. (Tainan, 2002), 235–240, Kluwer Acad. Publ., Dordrecht, 2003.Google Scholar