On a conjecture of Vukman

Abstract

ABSTRACT. 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 D R x R R and f z D(z, z) be a symmetric hi-derivation and its trace, respectively Suppose that f(x) E Z(R)for all x R, where .fk+l(X) [.fk(X),X] for k > 1 and f](x) f(x), thenD =0

SKEW n-DERIVATIONS ON SEMIPRIME RINGS

“…He extended classical Posner Theorem [28] to symmetric biderivations in prime and semiprime rings. Thereafter many literatures were focused on biderivations of prime and semiprime rings (see [1,2,3,4,5,6,9,10,12,13,14,16,17,19,20,25,30,32,33,34] for reference). Among these papers, the most important results are due to Brešar [8,9,10].…”

Symmetric bi-derivations and their generalizations on group algebras

“…for i > 1 and f 1 (x) = f (x). Deng [5] gave an affirmative answer to the Vukman's conjecture. For related results on symmetric bi-derivations on Banach algebras see [3,13]; see also [4,6,12] for study of generalized bi-derivations and Jordan bi-derivations.…”

“…Symmetric bi-derivations on rings have been introduced and studied by Maksa [9,10]. Several authors continued this investigations [2,5,[15][16][17][18]. For example, Vukman [16]…”