ON NILPOTENT POWER M R-GROUPS

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 275; no. 6; pp. 653 - 659
• Main Authors: Amaglobeli, M, Bokelavadze, T
• Format: Journal Article
• Language: English
• Published: Springer 02.10.2023
• ISSN: 1072-3374
• DOI: 10.1007/s10958-023-06706-5

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.