Explicit norm one elements for ring actions of finite abelian groups

It is known that the norm map N _G for the action of a finite groupG on a ringR is surjective if and only if for every elementary abelian subgroupU ofG the norm map N _U is surjective. Equivalently, there exists an elementx _G ∈R satisfying N _G (x _G )=1 if and only if for every elementary abelian subgroupU there exists an elementx _U ∈R such that N _U (x _U )=1. When the ringR is noncommutative, it is an open problem to find an explicit formula forx _G in terms of the elementsx _U . We solve this problem when the groupG is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case whenG is a cyclicp-group. Supported by TMR-Grant ERB FMRX-CT97-0100 of the European Union.