A basic functional identity with applications to Jordan σ-biderivations

Let R be a noncommutative prime ring with extended centroid C and maximal left ring of quotients Q(ml)(R). The aim of the paper is to study a basic functional identity concerning bi-additive maps on R. Precisely, it is proved that a bi-additive map B: R x R -> Q(ml)(R) satisfying [B(x,y),[x,y]] = 0 for all x,y is an element of R must be of the form (x,y) bar right arrow lambda[x,y] + mu(x,y) for x,y is an element of R, where lambda is an element of C and mu : R x R -> C is a bi-additive map. As applications to the theorem, Jordan sigma-biderivations with sigma an epimorphism and additive commuting maps on noncommutative Lie ideals of R are characterized.