### 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 |
---|---|

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

}

*Israel Journal of Mathematics*, vol. 175, no. 1, pp. 157-178. https://doi.org/10.1007/s11856-010-0007-z

**Higher derivations of ore extensions.** / Chuang, Chen Lian; Lee, Tsiu Kwen; Liu, Cheng Kai; Tsai, Yuan Tsung.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Higher derivations of ore extensions

AU - Chuang, Chen Lian

AU - Lee, Tsiu Kwen

AU - Liu, Cheng Kai

AU - Tsai, Yuan Tsung

PY - 2010/1/1

Y1 - 2010/1/1

N2 - 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 ζn ∈ CS(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.

AB - 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 ζn ∈ CS(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.

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U2 - 10.1007/s11856-010-0007-z

DO - 10.1007/s11856-010-0007-z

M3 - Article

AN - SCOPUS:77949931063

VL - 175

SP - 157

EP - 178

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -