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Turkish Journal of Mathematics

NURCAN ARGAÇ

DOI

-

Abstract

In this paper, we introduce the notion of two-sided \alpha-derivation of a near-ring and give some generalizations of [1]. Let N be a near ring. An additive mapping f: N\rightarrow N is called an { \it (\alpha, \beta)-derivation } if there exist functions \alpha,\beta : N\rightarrow N such that f(xy)=f(x)\alpha(y)+\beta (x)f(y) for all x,y\in N. An additive mapping d:N\rightarrow N is called a two-sided \alpha-derivation if d is an (\alpha,1)-derivation as well as a (1,\alpha)-derivation. The purpose of this paper is to prove the following two assertions: (i) Let N be a semiprime near-ring, I be a subset of N such that 0\in I, IN\subseteq I and d be a two-sided \alpha-derivation of N. If d acts as a homomorphism on I or as an anti-homomorphism on I under certain conditions on \alpha, then d(I)= {0}. (ii) Let N be a prime near-ring, I be a nonzero semigroup ideal of N, and d be a (\alpha, 1)-derivation on N. If d+d is additive on I, then (N,+) is abelian.

Keywords

Prime near-ring, semiprime near-ring, (\alpha, 1)-derivation, (1, \alpha)-derivation, two-sided \alpha-derivation.

First Page

195

Last Page

204

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