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.
Highlights
Throughout this paper, N is a left near-ring
An additive mapping d : R ! R is said to be a derivation if d(xy) = xd(y) + d(x)y for all x; y 2 N: An additive mapping f : R ! R is called a generalized derivation if there exists a derivation d : R ! R such that f = f (x)y + xd(y) for all x; y 2 R: Many papers in literature have investigated the commutativity of prime rings satisfying certain functional identities involving derivations or generalized derivations
They obtained the commutative rings 3 prime near-rings N satisfying some di¤erential identities where d is a multiplicative derivation of N: In the present paper, we shall prove these results for multiplicative generalized derivations of a 3 prime near-ring N: The results obtained in this paper extend, unify and complement several known results
Summary
Throughout this paper, N is a left near-ring. A near-ring N is called zero symmetric if 0x = 0 for all x 2 N (recall that left distributive yields x0 = 0): Z will represent the multiplicative center of N; that is Z = fx 2 N j yx = xy for all y 2 N g: A near-ring N is said to be 3 prime if xN y = f0g implies x = 0 or y = 0: For any x; y 2 N; as usual [x; y] = xy yx and xoy = xy + yx will denote the well-known Lie and Jordan product, respectively.
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