A Note on Polynomial Rings over Nil Rings

  • M. A. Chebotar
  • W. -F. Ke
  • P. -H. LeeEmail author
  • E. R. Puczyłowski
Part of the Trends in Mathematics book series (TM)


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 [8] and gives a new approximation, in that case, of a positive solution of Köthe’s problem.


Nil ring polynomial ring Brown-McCoy radical Köthe’s problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    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
  2. [2]
    Ferrero, M.; Wisbauer, R. Unitary strongly prime rings and related radicals. J. Pure Appl. Algebra 181 (2003), 209–226.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Krempa, J. Logical connections among some open problems in non-commutative rings. Fund. Math. 76 (1972), 121–130.MathSciNetGoogle Scholar
  4. [4]
    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
  5. [5]
    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
  6. [6]
    Puczyłowski, E.R.; Smoktunowicz, A. On maximal ideals and the Brown-McCoy radical of polynomial rings. Comm. Algebra 26 (1998), 2473–2482.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Smoktunowicz, A. On some results related to Köthe’s conjecture. Serdica Math. J. 27 (2001), 159–170.zbMATHMathSciNetGoogle Scholar
  8. [8]
    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

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • M. A. Chebotar
    • 1
  • W. -F. Ke
    • 2
  • P. -H. Lee
    • 3
    Email author
  • E. R. Puczyłowski
    • 4
  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA
  2. 2.Department of MathematicsNational Cheng Kung University and National Center for Theoretical Sciences, Tainan OfficeTainanTaiwan
  3. 3.Department of MathematicsNational Taiwan University and National Center for Theoretical Sciences, Taipei OfficeTaipeiTaiwan
  4. 4.Institute of MathematicsUniversity of WarsawWarsawPoland

Personalised recommendations