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We answer a question by Shestakov on the Jacobson radical in differentialpolynomial rings. We show that if R is a locally nilpotent ring with aderivation D then R[X;D] need not be Jacobson radical. We also show thatJ(R[X;D])\cap R is a nil ideal of R in the case where D is a locally nilpotentderivation and R is an algebra over an uncountable field.

Differential poynomial rings over locally nilpotent rings need not be Jacobson radical