Let
R be a prime ring and
D a nonzero derivation of
R. If one of the four conditions holds in
R, then
R is commutative:
| (i) |
X
2D(X)−D(X)X2∈Z(R), CharR≠2;
|
| (ii) |
X
2D(X)−XD(X)X∈Z(R), CharR≠2;
|
| (iii) |
X
3D(X)−D(X)X3=0, CharR≠2,3;
|
| (iv) |
X
nD(X)+XmD(X)Xn−m∈Z(R), wherem, n are fixed integers,0≤m≤n and CharR=0 or CharR>n.
|
1980 Mathematics Subject Classification (1985 Revision) Primary 16A12 - 16A68 - 16A72
Key words and phrases Prime ring - derivation - commutativity