On generalaized derivations with commutativity of semiprime rings

• Published in: Journal of Basrah Researches : Sciences. Vol. 37; no. 4C; pp. 132 - 133
• Main Authors: Atiyyah, Muhsin Jabal, Resan, Dalal Ibrahim
• Format: Journal Article
• Language: English
• Published: Basrah, Iraq University of Basrah, College of Education for Pure Sciences 2011
• Subjects: Algebra, Abstract; Noncommutative rings; Rings (Algebra); الجبر التجريدي; الحقول الجبرية
• ISSN: 1817-2695

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.