Strong commutativity preserving maps on rings

Suppose is a unital ring having an idempotent element e which satisfies ae=0 implies $a=0$ and a(1-e)=0 implies $a=0$. In this paper, we aim to characterize the map f:→, f is surjective and $[f(x),f(y)]=[x,y]$ for all x,y∈. It is shown that f(x)=α x +ξ (x) for all x∈, where α ∈ (), α²=1, and ξ is a map from into (). As an application, a characterization of nonlinear surjective maps preserving strong commutativity on von Neumann algebras with no central summands of type I₁ is obtained.