Addition Theorems and Group Rings

We establish several addition theorems on finite abelian groups by employing a group ring as a useful tool. Among several results the following is proved. Letpbe a prime, and letG=Zpe1⊗…⊗Zpenwith 1⩽e1⩽…⩽en. Putw=(1/(pen−1))×∑ni=1 (pei−1). Then, for anyt⩾(pen−1) log w+pen−2 linear basesB1, …, BtofGtheir union (with repetitions) ∪ti=1 Biforms an additive basis ofG.