Glasnik Matematicki, Vol. 39, No.1 (2004), 27-30.

ON GENERALIZED DERIVATIONS AS HOMOMORPHISMS AND ANTI-HOMOMORPHISMS

Nadeem-ur-Rehman

Department of Mathematics, Birla Institute of Technology and Science, Pilani 333031, Rajasthan, India
e-mail: rehman100@postmark.net


Abstract.   The concept of derivations as well as generalized derivations (i.e. Ia,b(x) = ax + xb, for all a,b R) have been generalized as an additive function F : R R satisfying F(xy) = F(x)y + xd(y) for all x,y R, where d is a nonzero derivation on R. Such a function F is said to be a generalized derivation. In the present paper it is shown that: if R is 2-torsion free prime ring, I ≠ 0 an ideal of R and F a generalized derivation of R such that either F(xy) = F(x)F(y) or F(xy) = F(y)F(x) for all x,y I, then R is commutative.

2000 Mathematics Subject Classification.   16W25, 16N60, 16U80.

Key words and phrases.   Prime rings, generalized derivations, torsion free rings, homomorphisms and anti-homomorphisms.


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