### Some Commutativity Theorems for Near-Rings with Left Multipliers

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**Authors:**A. BOUA, A. Y. ABDELWANIS AND A. CHILLALI

**DOI:**10.46793/KgJMat2002.205B

**Abstract:**

Let ???? be a 3-prime near-ring with the center Z(????), and U be a nonzero semigroup ideal of ????. In the present paper it is shown that a 3-prime near-ring ???? is a commutative ring if and only if it admits left multipliers ℱ and G satisfying any one of the following properties: (i)ℱ(x)G(y) ± [x,y] ∈ Z(????); (ii)ℱ(x)G(y) ± x ∘ y ∈ Z(????); (iii)ℱ(x)G(y) ± yx ∈ Z(????); (iv)ℱ(x)G(y) ± xy ∈ Z(????) and (v)ℱ([x,y]) ± G(x ∘ y) ∈ Z(????) for all x,y ∈ U.

**Keywords:**

3-Prime near-ring, derivations, commutativity, generalized derivation, left multiplier.

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