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

Shûichi Ikehata

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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 = 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.

Original language English 145-149 5 International Journal of Pure and Applied Mathematics 50 1 Published - 2009

### Fingerprint

Skew Polynomial Ring
Semiprime Ring
Polynomials
If and only if
Ring
Polynomial

### Keywords

• Derivation
• Separable polynomial
• Skew polynomial ring

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

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

In: International Journal of Pure and Applied Mathematics, Vol. 50, No. 1, 2009, p. 145-149.

Research output: Contribution to journalArticle

@article{35437c4aa480401c8c8fa01b5de6a264,
title = "A note on separable polynomials of degree 3 in skew polynomial rings",
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 = 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.",
keywords = "Derivation, Separable polynomial, Skew polynomial ring",
author = "Sh{\^u}ichi Ikehata",
year = "2009",
language = "English",
volume = "50",
pages = "145--149",
journal = "International Journal of Pure and Applied Mathematics",
issn = "1311-8080",
number = "1",

}

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 -