Derivations in prime near-rings

Source abstract

Let N be a prime near-ring with center Z . The purpose of this paper is to study derivations on N . We show two main results: (1) Let N be 2-torsion-free, and let D 1 {D_1} and D 2 {D_2} be derivations on N such that D 1 D 2 {D_1}{D_2} is also a derivation. Then either D 1 {D_1} or D 2 {D_2} is zero if and only if [ D 1 ( x ) , D 2 ( y ) ] = 0 [{D_1}(x),{D_2}(y)] = 0 for all x , y ∈ N x,y \in N . (2) Let n be an integer ≥ 1 \geq 1 , N be n !-torsion-free, and D a derivation with D n ( N ) = { 0 } {D^n}(N) = \{ 0\} . Then D ( Z ) = { 0 } D(Z) = \{ 0\} .