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