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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - M. G. Amaglobeli
AU  - A. G. Myasnikov
AU  - T. T. Nadiradze
TI  - Varieties of exponential $R$-groups
JO  - Algebra i logika
PY  - 2023
SP  - 179
EP  - 204
VL  - 62
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2023_62_2_a1/
LA  - ru
ID  - AL_2023_62_2_a1
ER  - 
%0 Journal Article
%A M. G. Amaglobeli
%A A. G. Myasnikov
%A T. T. Nadiradze
%T Varieties of exponential $R$-groups
%J Algebra i logika
%D 2023
%P 179-204
%V 62
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2023_62_2_a1/
%G ru
%F AL_2023_62_2_a1
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/

[1] R. Lyndon, “Groups with parametric exponents”, Trans. Am. Math. Soc., 96 (1960), 518–533 | DOI | MR | Zbl

[2] A. G. Myasnikov, V. N. Remeslennikov, “Stepennye gruppy. I: osnovy teorii i tenzornye popolneniya”, Sib. matem. zh., 35:5 (1994), 1106–1118 | MR | Zbl

[3] M. G. Amaglobeli, V. N. Remeslennikov, “Svobodnye 2-stupenno nilpotentnye $R$-gruppy”, Dokl. RAN, 443:4 (2012), 410–413 | Zbl

[4] M. G. Amaglobeli, “Funktor tenzornogo popolneniya v kategoriyakh stepennykh $MR$-grupp”, Algebra i logika, 57:2 (2018), 137–148 | MR | Zbl

[5] A. G. Myasnikov, V. N. Remeslennikov, “Exponential groups. II: Extensions of centralizers and tensor completion of CSA-groups”, Int. J. Algebra Comput., 6:6 (1996), 687–711 | DOI | MR | Zbl

[6] G. Baumslag, A. Myasnikov, V. Remeslennikov, “Discriminating completions of hyperbolic groups”, Geom. Dedicata, 92 (2002), 115–143 | DOI | MR | Zbl

[7] M. G. Amaglobeli, V. N. Remeslennikov, “Rasshirenie tsentralizatora v nilpotentnykh gruppakh”, Sib. matem. zh., 54:1 (2013), 8–19 | MR | Zbl

[8] M. Amaglobeli, V. Remeslennikov, “Algorithmic problems for class-2 nilpotents $M\!R$-groups”, Georgian Math. J., 22:4 (2015), 441–449 | DOI | MR | Zbl

[9] M. G. Amaglobeli, “Mnogoobraziya stepennykh $M\!R$-grupp”, Dokl. RAN. Matem., inform., prots. upr., 490 (2020), 5–8 | DOI | Zbl

[10] G. Baumslag, “Free abelian $X$-groups”, Ill. J. Math., 30:2 (1986), 235–245 | MR | Zbl

[11] V. A. Gorbunov, Algebraicheskaya teoriya kvazimnogoobrazii, Sibirskaya shkola algebry i logiki, Nauch. kniga, Novosibirsk, 1999

[12] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, SPb, Lan, 2009 | MR

[13] M. Kholl, Teoriya grupp, IL, M., 1962