On generalized derivations of prime near-rings

Abstract

Let N be a 2-torsion free prime near-ring with center Z, (f; d) and (g, h) two generalized derivations on N. In this case: (i) If f([x; y]) = 0 or f([x; y]) = $\pm$[x; y] or $f^2(x)\in Z$ for all x; y $\in$ N, then N is a commutative ring. (ii) If a $\in$ N and [f(x); a] = 0 for all x $\in$ N, then d(a) $\in$ Z. (iii) If (fg; dh) acts as a generalized derivation on N, then f = 0 or g = 0.

Let N be a 2-torsion free prime near-ring with center Z, (f; d) and (g, h) two generalized derivations on N. In this case: (i) If f([x; y]) = 0 or f([x; y]) = $\pm$[x; y] or $f^2(x)\in Z$ for all x; y $\in$ N, then N is a commutative ring. (ii) If a $\in$ N and [f(x); a] = 0 for all x $\in$ N, then d(a) $\in$ Z. (iii) If (fg; dh) acts as a generalized derivation on N, then f = 0 or g = 0.