ON COMMUTATIVITY OF PRIME NEAR-RINGS WITH MULTIPLICATIVE GENERALIZED DERIVATION

Abstract

In the present paper, we shall prove that 3 near-ring N is commutative ring, if any one of the following conditions are satis.ed: (i) f(N) subset of Z; (ii) f([x; y]) = 0; (iii) f([x; y]) = +/-[x; y]; (iv) f([x; y]) = +/-(xoy); (v) f([x; y]) = [f(x); y]; (vi) f([x; y]) = [x; f(y)]; (vii) f([x; y]) = [d(x); y]; (viii) f([x; y]) = d(x)oy;(ix) [f(x); y] is an element of Z for all x; y is an element of N where f is a nonzero multiplicative generalized derivation of N associated with a multiplicative derivation d.