IDENTITIES IN 3-PRIME NEAR-RINGS WITH LEFT MULTIPLIERS

Let N be a 3-prime near-ring with the center Z(N ) and n ≥ 1 be a fixed positive integer. In the present paper it is shown that a 3-prime near-ring N is a commutative ring if and only if it admits a left multiplier F satisfying any one of the following properties: (i) F n([x, y]) ∈ Z(N ), (ii) F n(x ◦ y) ∈ Z(N ), (iii)F n([x, y])±(x◦ y) ∈ Z(N ) and (iv)F n([x, y])±x◦ y ∈ Z(N ), for all x, y ∈ N .