On a theorem of J. Vukman

Abstract

LetR be a ring. A bi-additive symmetric mappingD:R × R → R is called a symmetric bi-derivation if, for any fixedy ∈ R, the mappingx ↦ D(x, y) is a derivation. J. Vukman [2, Theorem 2] proved that, ifR is a non-commutative prime ring of characteristic not two and three, and ifD:R × R → R is a symmetric bi-derivation such that [D(x, x), x] lies in the center ofR for allx ∈ R, thenD = 0. This result is in the spirit of the well-known theorem of Posner [1, Theorem 2], which states that the existence of a nonzero derivationd on a prime ringR, such that [d(x), x] lies in the center ofR for allx ∈ R, forcesR to be commutative. In this paper we generalize the result of J. Vukman mentioned above to nonzero two-sided ideals of prime rings of characteristic not two and we prove the following.