Products of Generalized Semiderivations of Prime Near Rings

Let N be a near ring. An additive mapping F : N -> N is said to be a generalized semiderivation on N if there exists a semiderivation d : N -> N associated with a function g : N -> N such that F(xy) = F(x) y + g(x) d(y) = d(x) g(y) + xF(y) and F(g(x)) = g(F(x)) for all x, y epsilon N. The purpose of the present paper is to prove some theorems in the setting of semigroup ideal of a 3-prime near ring admitting a pair of suitably-constrained generalized semiderivations, thereby extending some known results on derivations and generalized derivations. We show that if N is 2-torsion free and F-1 and F-2 are generalized semiderivations such that F1F2 = 0, then F-1 = 0 or F-2 = 0; we prove other theorems asserting triviality of F-1 or F-2; and we also prove some commutativity theorems.