Let K be a commutative ring with a unit element 1. Let Γ be a finite group acting on K via a map t: Γ→Aut(K). For every subgroup H≤Γ define tr H :K→K H by tr h (x)=Σσ∈H σ(x). We prove Theorem: trΓ is surjective onto K Γ if and only if tr P is surjective onto K P for every (cyclic) prime order subgroup P of Γ. This is false for certain non-commutative rings K.