Generalized Derivations on Prime Near Rings
Let be a near ring. An additive mapping is said to be a
right generalized (resp., left generalized) derivation with associated derivation on if (resp., ) for all . A mapping
 is said to be a generalized derivation with associated derivation on if
 is both a right generalized and a left generalized derivation with associated derivation
 on . The purpose of the present paper is to prove some theorems in the setting of
a semigroup ideal of a 3-prime near ring admitting a generalized derivation, thereby
extending some known results on derivations.
ON GENERALIZED n-DERIVATIONS IN NEAR-RINGS
In this paper, we introduce the notion of generalized n-derivation in near-ring N and investigate several identities involving generalized n-derivations of a prime near-ring N which force N to be a commutative ring. Finally some more related results are also obtained.
Posner and herstein theorems for derivations of 3-prime near-rings
The analog of Posner's theorem on the composition of two derivations in prime rings is proved for 3-prime near-rings. It is shown that if d is a nonzero derivation of a 2-torsionfree 3-prime near-ring N and an element a ∊ N is such that axd = xda for all x ∊ N, then a is a central element. As a consequence it is shown that if d\ and d2 are nonzero derivations of a 2-torsionfree 3-prime near-ring N and xd1yd2 = yd2xd1 for all x, y ∊ N, then N is a commutative ring. Thus two theorems of Herstein are generalized