### Abstract

Let R be a prime ring and δ a derivation of R. Divided powers, of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that, is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form, where ζ_{n} ∈ C_{S}(R), the centralizer of R in S. As an application, we also use the Ore extension R[x; δ] to deduce Kharchenko's theorem for a single derivation. These results are extended to the Ore extension R[X;D] of R by a sequence D of derivations of R.

Original language | English |
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Pages (from-to) | 157-178 |

Number of pages | 22 |

Journal | Israel Journal of Mathematics |

Volume | 175 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2010 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*175*(1), 157-178. https://doi.org/10.1007/s11856-010-0007-z