#### 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.
*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.

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