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.