On the semiprimitivity of skew polynomial rings. (Q2462248)

Let \(R\) be a ring with identity, \(\alpha\) an injective endomorphism of \(R\), which is not assumed to be surjective, and \(\delta\) an \(\alpha\)-derivation of \(R\). \textit{S. A. Amitsur} [Can. J. Math. 8, 355-361 (1956; Zbl 0072.02404)] has shown that, if \(R\) has no nil ideal then the polynomial ring \(R[x]\) is semiprimitive. This result was extended to skew polynomial rings of the form \(R[x;\alpha,\delta]\) by many authors. \textit{A. El Ahmar} [Arch. Math. 32, 13-15 (1979; Zbl 0398.16005)] has shown that if \(R\) is semiprime Noetherian and \(\alpha\) is a monomorphism, then \(R[x;\alpha]\) is semiprimitive. \textit{A. Moussavi} [Proc. Edinb. Math. Soc., II. Ser. 36, No. 2, 169-178 (1993; Zbl 0804.16029)] has extended this result to the skew polynomial ring \(R[x;\alpha,\delta]\); \textit{A. D. Bell} [Commun. Algebra 13, 1743-1762 (1985; Zbl 0567.16002)] has proved that if \(R\) is semiprime left Goldie with \(\alpha\) an automorphism and \(\delta\) an \(\alpha\)-derivation, then \(R[x;\alpha,\delta]\) is semiprimitive left Goldie. He has also commented that it is not known whether this generalizes to the case where \(\alpha\) is not assumed to be surjective. In this paper the authors give an affirmative answer to Bell's question.