On nilpotent power $MR$-groups

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.