### Abstract

Let B be a ring with identity 1, Z the center of B, D a derivation of B, and B[X; D] the skew polynomial ring such that αX = Xα + D(α) for each α ε B. Assume that 3 = 0 and Z is a semiprime ring. Let f = X^{3} -Xa -b ε B[X; D] such that f B[X; D] = B[X; D] f. Then we prove that f is a separable polynomial in B[X; D] if and only if there exits an element z in Z such that D^{2}(z) -za = 1.

Original language | English |
---|---|

Pages (from-to) | 145-149 |

Number of pages | 5 |

Journal | International Journal of Pure and Applied Mathematics |

Volume | 50 |

Issue number | 1 |

Publication status | Published - 2009 |

### Fingerprint

### Keywords

- Derivation
- Separable polynomial
- Skew polynomial ring

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*International Journal of Pure and Applied Mathematics*,

*50*(1), 145-149.

**A note on separable polynomials of degree 3 in skew polynomial rings.** / Ikehata, Shûichi.

Research output: Contribution to journal › Article

*International Journal of Pure and Applied Mathematics*, vol. 50, no. 1, pp. 145-149.

}

TY - JOUR

T1 - A note on separable polynomials of degree 3 in skew polynomial rings

AU - Ikehata, Shûichi

PY - 2009

Y1 - 2009

N2 - Let B be a ring with identity 1, Z the center of B, D a derivation of B, and B[X; D] the skew polynomial ring such that αX = Xα + D(α) for each α ε B. Assume that 3 = 0 and Z is a semiprime ring. Let f = X3 -Xa -b ε B[X; D] such that f B[X; D] = B[X; D] f. Then we prove that f is a separable polynomial in B[X; D] if and only if there exits an element z in Z such that D2(z) -za = 1.

AB - Let B be a ring with identity 1, Z the center of B, D a derivation of B, and B[X; D] the skew polynomial ring such that αX = Xα + D(α) for each α ε B. Assume that 3 = 0 and Z is a semiprime ring. Let f = X3 -Xa -b ε B[X; D] such that f B[X; D] = B[X; D] f. Then we prove that f is a separable polynomial in B[X; D] if and only if there exits an element z in Z such that D2(z) -za = 1.

KW - Derivation

KW - Separable polynomial

KW - Skew polynomial ring

UR - http://www.scopus.com/inward/record.url?scp=78649767927&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78649767927&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:78649767927

VL - 50

SP - 145

EP - 149

JO - International Journal of Pure and Applied Mathematics

JF - International Journal of Pure and Applied Mathematics

SN - 1311-8080

IS - 1

ER -