ON NILPOTENT POWER M R-GROUPS - Amaglobeli, M; Bokelavadze, T (Journal of mathematical sciences (New York, N.Y.), 2023)
The notion of a power M R-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 M R-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 M R-groups are introduced. Three versions of the definition of nilpotent power M R-groups of step n are discussed. We prove that all these definitions are equivalent for n = 1, 2. The question on the coincidence of these notions for n > 2 remains open. Moreover, we prove that the tensor completion of a 2-step nilpotent M R-group is 2-step nilpotent.