Differential polynomial rings over locally nilpotent rings need not be Jacobson radical. (Q403607)

In 2011 I. P. Shestakov asked the following question. Let \(R\) be a locally nilpotent associative ring with a derivation \(D\) and \(S=R[X;D]\) a skew polynomial ring. Is the Jacobson radical of \(S\) equal to \(S\)? It follows from results by \textit{M. Ferrero} et al. [J. Lond. Math. Soc., II. Ser. 28, 8-16 (1983; Zbl 0518.16003)] that the problem has positive answer if \(R\) is a commutative algebra over a field of characteristic zero. The main result shows that in general Shestakov's problem has a negative answer. If \(R\) is any algebra over an uncountable field then \(J(R[X;D])\cup R\) is nil provided \(D\) is locally nilpotent.