Products of Generalized Semiderivations of Prime Near Rings

Abstract

Let N be a near ring. An additive mapping $F:N\longrightarrow N$ is said to be a generalized semiderivation on N if there exists a semiderivation $d:N\longrightarrow N$ associated with a function $g:N\longrightarrow N$ 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 $x,y \in N$. 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 $F_1F_2=0$, 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.

First author is supported by a grant from Science and Engineering Research Board (SERB), DST, New Delhi, India. Grant No. SR/S4/MS:852/13.