Symmetric bi-derivations on prime and semi-prime rings

Summary

LetR be a ring. A bi-additive symmetric mappingD(.,.): R × R → R is called a symmetric bi-derivation if, for any fixedy ∈ R, a mappingx ↦ D(x, y) is a derivation. The purpose of this paper is to prove some results concerning symmetric bi-derivations on prime and semi-prime rings. We prove that the existence of a nonzero symmetric bi-derivationD(.,.): R × R → R, whereR is a prime ring of characteristic not two, with the propertyD(x, x)x = xD(x, x), x ∈ R, forcesR to be commutative. A theorem in the spirit of a classical result first proved by E. Posner, which states that, ifR is a prime ring of characteristic not two andD ₁,D ₂ are nonzero derivations onR, then the mappingx ↦ D ₁(D ₂ (x)) cannot be a derivation, is also presented.