Geodesic
Varieties of exponential $R$-groups
Algebra i logika, Tome 62 (2023) no. 2, pp. 179-204

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The notion of an exponential $R$-group, where $R$ is an arbitrary associative ring with unity, was introduced by R. Lyndon. Myasnikov and Remeslennikov refined the notion of an $R$-group by introducing an additional axiom. In particular, the new concept of an exponential $M R$-group ($R$-ring) is a direct generalization of the concept of an $R$-module to the case of noncommutative groups. We come up with the notions of a variety of $M R$-groups and of tensor completions of groups in varieties. Abelian varieties of $M R$-groups are described, and various definitions of nilpotency in this category are compared. It turns out that the completion of a $2$-step nilpotent $M R$-group is $2$-step nilpotent.
Keywords: Lyndon's $R$-group, varietiy of $M R$-groups, $\alpha$-commutator, nilpotent $M R$-group, tensor completion.
Mots-clés : $M R$-group, $R$-commutant 
@article{AL_2023_62_2_a1,
     author = {M. G. Amaglobeli and A. G. Myasnikov and T. T. Nadiradze},
     title = {Varieties of exponential $R$-groups},
     journal = {Algebra i logika},
     pages = {179--204},
     publisher = {mathdoc},
     volume = {62},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2023_62_2_a1/}
}
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M. G. Amaglobeli; A. G. Myasnikov; T. T. Nadiradze. Varieties of exponential $R$-groups. Algebra i logika, Tome 62 (2023) no. 2, pp. 179-204. http://geodesic.mathdoc.fr/item/AL_2023_62_2_a1/