Nilpotency of the multiplicative group of a group ring

It is proven that if K is a commutative ring of characteristic p^m while group G contains p-elements, then the multiplicative group UKG of group ring KG is nilpotent if and only if G is nilpotent and its commutant G is a finite p-group. Those group algebras KG are described for which the nilpotency classes of groups G and UKG coincide.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 191–200, February, 1972.In conclusion, the author wishes to express her gratitude to A. A. Bovdi for his scientific direction.