Symmetric n-Additive Mappings Admitting Semiprime Ring
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Authors: K. KUMAR
DOI: 10.46793/KgJMat2505.755K
Abstract:
Let ℛ be a ring with centre Z(ℛ). An n-additive map D : ℛn →ℛ is called symmetric n-additive if D(x1,…,xn) = D(xπ(1),…,xπ(n)) for all xi ∈ℛ and for every permutation (π(1),π(2),…,π(n)). A mapping △ : ℛ→ℛ defined by △(x) = D(x,x,…,x) is called the trace of D. In this paper, we prove that a nonzero Lie ideal L of a semiprime ring ℛ of characteristic different from (2n − 2) is central, if it satisfies any one of the following properties: (i) △([x,y]) ∓ xy ∈ Z(ℛ); (ii) △([x,y]) ∓ [y,x] ∈ Z(ℛ); (iii) △(xy) ∓△(x) ∓ [x,y] ∈ Z(ℛ); (iv) △([x,y]) ∓ yx ∈ Z(ℛ); (v) △(xy) ∓△(y) ∓ [x,y] ∈ Z(ℛ).
Keywords:
Semiprime rings, Lie ideals, Symmetric n-additive mapping, Trace.
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