IJPAM: Volume 70, No. 7 (2011)

ON MULTILINEAR POLYNOMIALS IN
PRIME RINGS WITH DERIVATIONS

Basudeb Dhara$^1$, Deepankar Das$^2$
$^1$Department of Mathematics
Belda College
Belda, Paschim Medinipur, 721424, W.B., INDIA
$^2$Department of Mathematics
Haldia Goverment College
Haldia, Purba Medinipur, 721657. W.B., INDIA


Abstract. Let $R$ be a prime ring with extended centroid $C$ and characteristic different from $2$, $d$ a nonzero derivation of $R$, $f(x_1,\ldots,x_n)$ a nonzero multilinear polynomial over $C$ and $I$ a nonzero right ideal of $R$. If the mapping $x\mapsto [d^2(x),d(x)]$ is centralizing on $\{f(x_1,\ldots,x_n)\vert x_1,\ldots,x_n\in I\}$, then one of the following holds:


(1) $d$ is inner derivation induced by an element $a\in Q$ such that $aI=(0)=a^2I$;


(2) $[[f(x_1,\ldots,x_n),x_{n+1}]x_{n+2}]=0$ is an identity for $I$.

Received: April 3, 2011

AMS Subject Classification: 16W25, 16R50, 16N60

Key Words and Phrases: prime ring, derivation, extended centroid, Martindale quotient ring

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 70
Issue: 7