Traces of Permuting n-Additive Maps and Permuting n-Derivations of Rings |
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Authors: | Mohammad Ashraf Malik Rashid Jamal |
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Institution: | 1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India
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Abstract: | Let R be a prime ring with center Z(R). For a fixed positive integer n, a permuting n-additive map ${\Delta : R^n \to R}$ is known to be permuting n-derivation if ${\Delta(x_1, x_2, \ldots, x_i x'_{i},\ldots, x_n) = \Delta(x_1, x_2, \ldots, x_i, \ldots, x_n)x'_i + x_i \Delta(x_1, x_2, \ldots, x'_i, \ldots, x_n)}$ holds for all ${x_i, x'_i \in R}$ . A mapping ${\delta : R \to R}$ defined by δ(x) = Δ(x, x, . . . ,x) for all ${x \in R}$ is said to be the trace of Δ. In the present paper, we have proved that a ring R is commutative if there exists a permuting n-additive map ${\Delta : R^n \to R}$ such that ${xy + \delta(xy) = yx + \delta(yx), xy- \delta(xy) = yx - \delta(yx), xy - yx = \delta(x) \pm \delta(y)}$ and ${xy + yx = \delta(x) \pm \delta(y)}$ holds for all ${x, y \in R}$ . Further, we have proved that if R is a prime ring with suitable torsion restriction then R is commutative if there exist non-zero permuting n-derivations Δ1 and Δ2 from ${R^n \to R}$ such that Δ1(δ 2(x), x, . . . ,x) = 0 for all ${x \in R,}$ where δ 2 is the trace of Δ2. Finally, it is shown that in a prime ring R of suitable torsion restriction, if ${\Delta_1, \Delta_2 : R^n \longrightarrow R}$ are non-zero permuting n-derivations with traces δ 1, δ 2, respectively, and ${B : R^n \longrightarrow R}$ is a permuting n-additive map with trace f such that δ 1 δ 2(x) = f(x) holds for all ${x \in R}$ , then R is commutative. |
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