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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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). https://doi.org/10.1007/s11856-017-1616-6