On generalaized derivations with commutativity of semiprime rings

الملخص EN

The main purpose of this paper is to study and investigate some results concerning generalized derivation D on semiprime ring R, we obtain R contains a non-zero central ideal and when D=0,then R is commutative.

This research has been motivated by the work of M.

Ashraf [1] and M.

A.

Quadri, M.

Shadab Khan and N.

Rehman[10].Throughout this paper, R will represent an associative ring with the center Z (R).We recall that R is semiprime if xRx =(0) implies x=o and it is prime if xRy=(o) implies x=o or y=o.A prime ring is semiprime but the converse is not true in general.

A ring R is 2-torsion free in case 2x = (o) implies that x=(o)for any x € R .

An additive mapping d:R→R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x,y € R.A mapping d is called centralizing if [d(x),x] € Z(R) for all x€ R, in particular , if [d(x),x] = o for all x € R, then it is called commuting, and is called central if d(x) Z(R) for all x € R.Every central mapping is obviously commuting but not conversely in general .

Following Bresar [3] an additive mapping D : R → R is called a generalized derivation on R if there exists a derivation d : R→R such that D(xy) = D(x)y+ xd(y) holds for all x,y € R.

However, generalized derivation covers the concept of derivation .

Also with d=o, a generalized derivation covers the concept of left multiplier (left centralizer) that is, an additive mapping D satisfying D(xy) = D(x)y for all x,y € R.

As usual, we write [x,y] for xy –yx and make use of the commutator identities [xy,z]=x[y,z]+[x,z]y and [x,yz]=y[x,z]+ [x,y]z.