Two results concerning symmetric bi-derivations on prime rings

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. The purpose of this paper is to prove two results concerning symmetric bi-derivations on prime rings. The first result states that, ifD1 andD2 are symmetric bi-derivations on a prime ring of characteristic different from two and three such thatD1(x, x)D2(x,x) = 0 holds for allx ∈ R, then eitherD1 = 0 orD2 = 0. The second result proves that the existence of a nonzero symmetric bi-derivation on a prime ring of characteristic different from two and three, such that [[D(x, x),x],x] ∈ Z(R) holds for allx ∈ R, whereZ(R) denotes the center ofR, forcesR to be commutative.