IJPAM: Volume 5, No. 4 (2003)

DERIVATIONS ON SEMIPRIME RINGS

M.S. Samman$^1$, A.B. Thaheem$^2$
Department of Mathematical Sciences
King Fahd University of Petroleum and Minerals
Dhahran 31261, SAUDI ARABIA
$^1$e-mail: msamman@kfupm.edu.sa
$^2$e-mail: athaheem@kfupm.edu.sa


Abstract.In this paper we investigate some properties of derivations on prime and semiprime rings. Among other results we prove that if $R$ is a semiprime ring, $I$ is a nonzero two-sided ideal of $R$ and $f,g$ are derivations of $R$ satisfying $f(x)y + yg(x) = 0$ for all $x,y\in I$, then $f(u)[x,y]
= [x,y]g(u) = 0$ for all $x,y\in I$; in particular, $f$ and $g$ map $I$ into the center of $R$. If $R$ is a noncommutative prime ring, then $f = g = 0$ on $R$, which may be regarded as an analog of Posner Lemma for a pair of derivations satisfying this identity.

Received: March 3, 2003

AMS Subject Classification: 16A70, 16N60, 16W25

Key Words and Phrases: prime ring, semiprime ring, commuting map, centralizing map, derivation, semi-commuting map, skew-commuting map

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2003
Volume: 5
Issue: 4