Geodesic
On nilpotent power $MR$-groups
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 3-9.

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

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.
Keywords: Lyndon $R$-group, Hall $R$-group, $MR$-group, $\alpha$-commutator, tensor completion, nilpotent $MR$-group. 
@article{INTO_2020_177_a0,
     author = {M. G. Amaglobeli and T. Bokelavadze},
     title = {On nilpotent power $MR$-groups},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {3--9},
     publisher = {mathdoc},
     volume = {177},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_177_a0/}
}
TY  - JOUR
AU  - M. G. Amaglobeli
AU  - T. Bokelavadze
TI  - On nilpotent power $MR$-groups
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 3
EP  - 9
VL  - 177
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_177_a0/
LA  - ru
ID  - INTO_2020_177_a0
ER  - 
%0 Journal Article
%A M. G. Amaglobeli
%A T. Bokelavadze
%T On nilpotent power $MR$-groups
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 3-9
%V 177
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_177_a0/
%G ru
%F INTO_2020_177_a0
M. G. Amaglobeli; T. Bokelavadze. On nilpotent power $MR$-groups. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 3-9. http://geodesic.mathdoc.fr/item/INTO_2020_177_a0/

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

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

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

[13] Amaglobeli M. G., Remeslennikov V. N., “Osnovy teorii mnogoobrazii nilpotentnykh $\mathrm{MR}$-grupp”, Sib. mat. zh., 57:6 (2016), 1197–1207 | MR | Zbl

[14] Myasnikov A. G., Remeslennikov V. N., “Formulnost mnozhestva maltsevskikh baz i elementarnye svoistva konechnomernykh algebr, II”, Sib. mat. zh., 24:2 (1983), 97–113 | MR | Zbl

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

[16] Amaglobeli M. G., “On the permutability of a functor of tensor completion with principal group operations”, Appl. Math. Inform. Mech., 15:1 (2010), 3–10 | MR

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

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

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

[20] Hall M., Jr., The Theory of Groups, Chelsea Publ., New York, 1976 | MR | Zbl

[21] Hall P., The Edmonton Notes on Nilpotent Groups, Queen Mary College, London, 1969 | MR

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

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

[24] Neumann H., Varieties of Groups, Springer-Verlag, New York, 1967 | MR | Zbl