IJPAM: Volume 50, No. 1 (2009)

A NOTE ON SEPARABLE POLYNOMIALS OF DEGREE $3$
IN SKEW POLYNOMIAL RINGS

Shûichi Ikehata
Department of Environmental and Mathematical Science
Faculty of Environmental Science and Technology
Okayama University
Tsushima, Okayama 700-8530, JAPAN
e-mail: [email protected]


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 $\alpha X = X\alpha + D(\alpha)$ for each $\alpha \in B$. Assume that $3 = 0$ and $Z$ is a semiprime ring. Let $f = X^3 - Xa - b \in B[X;D]$ such that $fB[X;D] = B[X;D]f$. 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 $D^2(z) - za =1$.

Received: December 26, 2008

AMS Subject Classification: 16S30, 16W20

Key Words and Phrases: separable polynomial, skew polynomial ring, derivation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 50
Issue: 1