Geodesic
On nilpotent power $MR$-groups
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 3-9

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The notion of a power $MR$-group, where $R$ is an arbitrary associative ring with unity, was introduced by R. Lyndon. A. G. Myasnikov and V. N. Remeslennikov gave a more precise definition of an $R$-group by introducing an additional axiom. In particular, the new notion of a power $MR$-group is a direct generalization of the notion of an $R$-module to the case of noncommutative groups. In the present paper, central series and series of commutants in $MR$-groups are introduced. Three variants of the definition of nilpotent power $MR$-groups of step $n$ are discussed. It is proved that, for $n=1,2$, all these definitions are equivalent. The question on the coincidence of these notions for $n>2$ remains open. Moreover, it is proved that the tensor completion of a 2-step nilpotent $MR$-group is 2-step nilpotent.
Keywords: Lyndon $R$-group, Hall $R$-group, $MR$-group, $\alpha$-commutator, tensor completion, nilpotent $MR$-group. 
@article{INTO_2020_177_a0,
     author = {M. G. Amaglobeli and T. Bokelavadze},
     title = {On nilpotent power $MR$-groups},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {3--9},
     publisher = {mathdoc},
     volume = {177},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_177_a0/}
}
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M. G. Amaglobeli; T. Bokelavadze. On nilpotent power $MR$-groups. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 3-9. http://geodesic.mathdoc.fr/item/INTO_2020_177_a0/