It is known that the norm map NG for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map NU is surjective. Equivalently, there exists an element xG ∈ R satisfying NG(xG) = 1 if and only if for every elementary abelian subgroup U there exists an element xU ∈ R such that NU(xU) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for xG in terms of the elements xU. We solve this problem when the group G is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case when G is a cyclic p-group.