On commutativity of prime near-rings with multiplicative generalized derivation

Abstract

In the present paper, we shall prove that 3-prime near-ring N is commutative ring, if any one of the following conditions are satisfied: (i) f(N)⊆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]∈Z for all x,y∈N where f is a nonzero multiplicative generalized derivation of N associated with a multiplicative derivation d.

Keywords

• Prime near-ring,derivation,multiplicative generalized derivation

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