Geodesic
Nilpotency of the multiplicative group of a group ring
Matematičeskie zametki, Tome 11 (1972) no. 2, pp. 191-200

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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. 
@article{MZM_1972_11_2_a8,
     author = {I. I. Khripta},
     title = {Nilpotency of the multiplicative group of a group ring},
     journal = {Matemati\v{c}eskie zametki},
     pages = {191--200},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_11_2_a8/}
}
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I. I. Khripta. Nilpotency of the multiplicative group of a group ring. Matematičeskie zametki, Tome 11 (1972) no. 2, pp. 191-200. http://geodesic.mathdoc.fr/item/MZM_1972_11_2_a8/