A note on the Jacobson radical of Ore extensions (Q6600737)

The authors present a simplified proof of the semiprimitivity of skew polynomial rings \(R[x; \alpha, \delta]\), demonstrating that the ring is semiprimitive and reduced if and only if \(R\) is \(\alpha\)-rigid. This proof unifies and extends several known results on Jacobson radicals, particularly for non-commutative rings. The findings apply to several important structures in quantum algebra, such as the quantum \(n\)-space \(O_u(S_n)\), the \(n\)th Weyl algebra \(A_n(S)\), and the quantized Weyl algebra \(A_u(S)\).\N\NThe article proves that these structures are semiprimitive when \(S\) is a reduced ring. The article determines the Jacobson radical of Ore extensions of NI rings (rings where the set of nilpotent elements forms an ideal), generalizing classical results. The article extends classical results on the Jacobson radical to more general classes of rings, particularly \(\alpha\)-rigid rings, and applies these results to non-commutative structures. The simplified proof of semiprimitivity is a notable improvement over earlier, more complex methods, making the results more accessible to readers.\N\NThe results offer an affirmative answer to the Köthe conjecture for a class of Ore extensions, under specific conditions on \(\alpha\) and \(\delta\). This is a significant contribution to the longstanding conjecture in ring theory and it is probably one of the most difficult open problem in ring theory. The Köthe conjecture suggests that if a ring does not contain a nil ideal, it cannot contain a one-sided nil ideal either. The article affirms this conjecture for Ore extensions, showing that if the set of nilpotent elements in \(R\) forms a nil ideal, then \(R[x; \alpha, \delta]\) satisfies the Köthe conjecture. This result strengthens existing partial solutions for this longstanding problem in non-commutative algebra.