##
**Five solved problems on radicals of Ore extensions.**
*(English)*
Zbl 1450.16014

Let \(R\) be a ring which is necessary to be a commutative ring, let \(\sigma: R\rightarrow R\) is a ring homomorphism, and let \(\delta: R \rightarrow R\) is a \(\sigma\)-derivation of \(R\). Then the ring extension \(R[X;\sigma,\delta]\) is an Ore extension which also called a skew polynomial rings five problems-their solution.

In this paper, the authors solve five problems on the radical of Ore extensions. At the first step, the authors present a positive answer for the question of whether the ring extension \(R[X,\delta]\) is locally nilpotent when \(R\) is a prime radical ring with a derivation \(\delta\) in Theorem 2.3. Moreover, the authors also give a consequence of Theorem 2.3 described in Corollary 2.4. The next problem solved by the authors is related to the stability of the Jacobson radical in graded rings. The authors explain that the Jacobson radical of a ring \(R\) need not be \(\delta\)-stable. In Theorem 3.3., the authors state that \(\delta(J(R)) \subseteq J(R)\) if \(\delta\) is a grading-preserving derivation on \(R\), where \(R \oplus_{i=1}^{\infty} R_i\) be an algebra over a field \(K\) of characteristic zero. As the third solution, the authors give a positive answer to the conjecture of whether \(N(R[X;\delta])=I[X;\delta]\) for some ideal \(I\) in \(R\). The fourth solution is explained in Theorem 5.3 which describes that the natural extension of the multiplication from \(R[X;\sigma,\delta]\) to \(R[[X;\sigma,\delta]]\) is well defined, where \(R\) is an algebra, \(\sigma\) is an endomorphism, and \(\delta\) is \(\sigma\)-derivation which is locally nilpotent. This result also raises an open question which is explained in Question 5.4. The final problem is related to graded algebras, homogeneity and other related questions. The authors explain their solution in Proposition 6.1, Theorem 6.12, and Proposition 6.14. In Proposition 6.1, the authors explain that \(R\) is a Behrens radical, where \(R=\oplus_{i=0}^{\infty}\) is graded nil. The authors also raise 7 questions related to this result. In Theorem 6.12, the authors describe that if \(R[X;\delta]\) is a graded locally nilpotent, then \(R[X;\delta]\) is a locally nilpotent. In Proposition 6.14, the authors give a sufficient condition for \(I[X;\sigma]^+\) to be graded nil and a sufficient condition for \(J(R[X;\sigma])^+\) to be nil. Finally, this result motivates three open questions which are explained in Question 6.15, Question 6.16, and Question 6.17, respectively.

In this paper, the authors solve five problems on the radical of Ore extensions. At the first step, the authors present a positive answer for the question of whether the ring extension \(R[X,\delta]\) is locally nilpotent when \(R\) is a prime radical ring with a derivation \(\delta\) in Theorem 2.3. Moreover, the authors also give a consequence of Theorem 2.3 described in Corollary 2.4. The next problem solved by the authors is related to the stability of the Jacobson radical in graded rings. The authors explain that the Jacobson radical of a ring \(R\) need not be \(\delta\)-stable. In Theorem 3.3., the authors state that \(\delta(J(R)) \subseteq J(R)\) if \(\delta\) is a grading-preserving derivation on \(R\), where \(R \oplus_{i=1}^{\infty} R_i\) be an algebra over a field \(K\) of characteristic zero. As the third solution, the authors give a positive answer to the conjecture of whether \(N(R[X;\delta])=I[X;\delta]\) for some ideal \(I\) in \(R\). The fourth solution is explained in Theorem 5.3 which describes that the natural extension of the multiplication from \(R[X;\sigma,\delta]\) to \(R[[X;\sigma,\delta]]\) is well defined, where \(R\) is an algebra, \(\sigma\) is an endomorphism, and \(\delta\) is \(\sigma\)-derivation which is locally nilpotent. This result also raises an open question which is explained in Question 5.4. The final problem is related to graded algebras, homogeneity and other related questions. The authors explain their solution in Proposition 6.1, Theorem 6.12, and Proposition 6.14. In Proposition 6.1, the authors explain that \(R\) is a Behrens radical, where \(R=\oplus_{i=0}^{\infty}\) is graded nil. The authors also raise 7 questions related to this result. In Theorem 6.12, the authors describe that if \(R[X;\delta]\) is a graded locally nilpotent, then \(R[X;\delta]\) is a locally nilpotent. In Proposition 6.14, the authors give a sufficient condition for \(I[X;\sigma]^+\) to be graded nil and a sufficient condition for \(J(R[X;\sigma])^+\) to be nil. Finally, this result motivates three open questions which are explained in Question 6.15, Question 6.16, and Question 6.17, respectively.

Reviewer: Puguh Wahyu Prasetyo (Yogyakarta)

### MSC:

16N20 | Jacobson radical, quasimultiplication |

16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |

16N60 | Prime and semiprime associative rings |

16N80 | General radicals and associative rings |

16S32 | Rings of differential operators (associative algebraic aspects) |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

16W50 | Graded rings and modules (associative rings and algebras) |