Abstract
Let R be a ring A bi‐additive symmetric mapping d : R × R → R is called a symmetric bi‐derivation if, for any fixed y ∈ R, the mapping x → D(x, y) is a derivation. The purpose of this paper is to prove the following conjecture of Vukman.Let R be a noncommutative prime ring with suitable characteristic restrictions, and let D : R × R → R and f : x → D(x, x) be a symmetric bi‐derivation and its trace, respectively. Suppose that fn(x) ∈ Z(R) for all x ∈ R, where fk+1(x) = [fk(x), x] for k ≥ 1 and f1(x) = f(x), then D = 0.
Highlights
Throughout this paper, R will denote an associative ring with center Z(R)
Let R be a ring A bi-additive symmetric mapping d R R R is called a symmetric hi-derivation if, for any fixed y E R, the mapping z D(z, 1) is a derivation The purpose of this paper is to prove the following conjecture of Vukman
Symmetric if D(x, ) holds for all pairs x, y R A symmetric mapping is called a symmetric biderivation, if D(z + z) D(z, z) + D(y, z) and D(xy, z) D(:r, z)u + xD(u, z) are fulfilled for all z,/ R The mapping f :R R defined by f(x) D(z,x) is called the trace of the symmetric bi-derivation D, and obviously, f(z + 1) f(x) + f(y) + 2D(z, y) The concept of a symmetric biderivation was introduced by Gy Maksa in [1,2] Some recent results cnceming symmetric bi-derivations of prime tings can be found in Vukman [3,4]
Summary
Throughout this paper, R will denote an associative ring with center Z(R). We write [z,y] for xy- /x, and I, for the inner derivation deduced by a A mapping D R R R will be called, symmetric if D(x, ) holds for all pairs x, y R A symmetric mapping is called a symmetric biderivation, if D(z + z) D(z, z) + D(y, z) and D(xy, z) D(:r, z)u + xD(u, z) are fulfilled for all z,/ R The mapping f :R R defined by f(x) D(z,x) is called the trace of the symmetric bi-derivation D, and obviously, f(z + 1) f(x) + f(y) + 2D(z, y) The concept of a symmetric biderivation was introduced by Gy Maksa in [1,2] Some recent results cnceming symmetric bi-derivations of prime tings can be found in Vukman [3,4]. Let R be a ring A bi-additive symmetric mapping d R R R is called a symmetric hi-derivation if, for any fixed y E R, the mapping z D(z, 1) is a derivation The purpose of this paper is to prove the following conjecture of Vukman Let R be a noncommutative prime ring with suitable characteristic restrictions, and let
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