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On polynomial rings over nil rings in several variables and the central closure of prime nil rings


We prove that the ring of polynomials in several commuting indeterminates over a nil ring cannot be homomorphically mapped onto a ring with identity, i.e. it is Brown-McCoy radical. It answers a question posed by Puczyłowski and Smoktunowicz. We also show that the central closure of a prime nil ring cannot be a simple ring with identity, solving a problem due to Beidar.

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  1. S. A. Amitsur, Nil radicals. Historical notes and some new results, (1973), 47᾿5. Colloq. Math. Soc. Janos Bolyai, Vol. 6.

    MathSciNet  MATH  Google Scholar 

  2. K. I. Beidar, W. S. Martindale, III and A. V. Mikhalev, Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 196, Marcel Dekker, Inc., New York, 1996.

  3. M. A. Chebotar, On a problem by Beidar concerning the central closure, Linear Algebra Appl. 429 (2008), 835᾿40.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. A. Chebotar, W.-F. Ke, P.-H. Lee and E. R. Puczylowski, A note on polynomial rings over nil rings, in Modules and comodules, Trends Math., Birkhauser Verlag, Basel, 2008, pp. 169᾿72.

    Chapter  Google Scholar 

  5. M. Ferrero and R. Wisbauer, Unitary strongly prime rings and related radicals, J. Pure Appl. Algebra 181 (2003), 209᾿26.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. E. Goodman and J. O'Rourke (eds.), Handbook of discrete and computational geometry, second ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004.

    Google Scholar 

  7. A. KauCikas and R. Wisbauer, On strongly prime rings and ideals, Comm. Algebra 28 (2000), 5461᾿473.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Krempa, Logical connections between some open problems concerning nil rings, Fund. Math. 76 (1972), 121᾿30.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. R. Puczytowski, Some questions concerning radicals of associative rings, in Theory of radicals (Szekszárd, 1991), Colloq. Math. Soc. János Bolyai, Vol. 61, North-Holland, Amsterdam, 1993, pp. 209᾿27.

    Google Scholar 

  10. E. R. Puczytowski, Questions related to Koethe's nil ideal problem, in Algebra and its applications, Contemp. Math., Vol. 419, Amer. Math. Soc., Providence, RI, 2006, pp. 269᾿83.

    Article  MathSciNet  MATH  Google Scholar 

  11. E. R. Puczytowski and A. Smoktunowicz, On maximal ideals and the Brown-McCoy radical of polynomial rings, Comm. Algebra 26 (1998), 2473᾿482.

    Article  MathSciNet  MATH  Google Scholar 

  12. E. R. Puczytowski and R. Wiegandt, Kostia's contribution to radical theory and related topics, in Rings and nearrings, Walter de Gruyter, Berlin, 2007, pp. 121᾿57.

    Google Scholar 

  13. R. Schneider, Convex bodies: the Brunn Minkowski theory, expanded ed., Encyclopedia ol Mathematics and its Applications, Vol. 151, Cambridge University Press, Cambridge, 2014.

  14. A. Smoktunowicz, Polynomial rings over nil rings need not be nil, J. Algebra 233 (2000), 427᾿36.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Smoktunowicz, On some results related to Köthe's conjecture, Serdica Math. J. 27 (2001), 159᾿70.

    MathSciNet  MATH  Google Scholar 

  16. A. Smoktunowicz, R[x, y] is Brown-McCoy radical if R[x] is Jacobson radical, in Proceedings of the Third international Algebra Conference (Tainan, 2002), Kluwer Acad. Publ., Dordrecht, 2003, pp. 235᾿40.

    Chapter  Google Scholar 

  17. A. Smoktunowicz, Makar Limanov's conjecture on free subalgebras, Adv. Math. 222 (2009), 2107᾿116.

    Article  MathSciNet  MATH  Google Scholar 

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Chebotar, M., Ke, WF., Lee, PH. et al. On polynomial rings over nil rings in several variables and the central closure of prime nil rings. Isr. J. Math. 223, 309–322 (2018).

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