In this paper, we introduce the notion of two-sided a-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 (a, b)-derivation } if there exist functions a,b : N\rightarrow N such that f(xy)=f(x)a(y)+b (x)f(y) for all x,y\in N. An additive mapping d:N\rightarrow N is called a two-sided a-derivation if d is an (a,1)-derivation as well as a (1,a)-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 a-derivation of N. If d acts as a homomorphism on I or as an anti-homomorphism on I under certain conditions on a, then d(I)= {0}. (ii) Let N be a prime near-ring, I be a nonzero semigroup ideal of N, and d be a (a, 1)-derivation on N. If d+d is additive on I, then (N,+) is abelian.

In this paper, we introduce the notion of two-sided a-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 (a, b)-derivation } if there exist functions a,b : N\rightarrow N such that f(xy)=f(x)a(y)+b (x)f(y) for all x,y\in N. An additive mapping d:N\rightarrow N is called a two-sided a-derivation if d is an (a,1)-derivation as well as a (1,a)-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 a-derivation of N. If d acts as a homomorphism on I or as an anti-homomorphism on I under certain conditions on a, then d(I)= {0}. (ii) Let N be a prime near-ring, I be a nonzero semigroup ideal of N, and d be a (a, 1)-derivation on N. If d+d is additive on I, then (N,+) is abelian.