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

ON GENERALIZED DERIVATIONS AS HOMOMORPHISMS AND ANTI-HOMOMORPHISMS

Nadeem-ur-Rehman

Abstract.   The concept of derivations as well as generalized derivations (i.e. I_a,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.