Geodesic
Subgroups of the general linear group containing the elementary subgroup over a commutative ring extension of rank 2
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 31, Tome 455 (2017), pp. 209-225 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Let $R=\prod_{i\in I}F_i$ be a direct product of fields and let $S=R[\sqrt d]=\prod_{i\in I}F_i[\sqrt{d_i}]$ be a ring extension of rank 2 of $R$. The subgroups of the general linear group $\operatorname{GL}(2n,R)$, $n\geq3$ that contain the elementary group $E(n,S)$ are described. It is shown that for every such a subgroup $H$ there exists a unique ideal $A\unlhd R$ such that $$ E(n,S)E(2n,R,A)\leq H\leq N_{\operatorname{GL}(2n,R)}(E(n,S)E(2n,R,A)). $$ 
@article{ZNSL_2017_455_a15,
     author = {T. N. Hoi and N. H. T. Nhat},
     title = {Subgroups of the general linear group containing the elementary subgroup over a~commutative ring extension of rank~2},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {209--225},
     year = {2017},
     volume = {455},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_455_a15/}
}
TY  - JOUR
AU  - T. N. Hoi
AU  - N. H. T. Nhat
TI  - Subgroups of the general linear group containing the elementary subgroup over a commutative ring extension of rank 2
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2017
SP  - 209
EP  - 225
VL  - 455
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2017_455_a15/
LA  - en
ID  - ZNSL_2017_455_a15
ER  - 
%0 Journal Article
%A T. N. Hoi
%A N. H. T. Nhat
%T Subgroups of the general linear group containing the elementary subgroup over a commutative ring extension of rank 2
%J Zapiski Nauchnykh Seminarov POMI
%D 2017
%P 209-225
%V 455
%U http://geodesic.mathdoc.fr/item/ZNSL_2017_455_a15/
%G en
%F ZNSL_2017_455_a15
T. N. Hoi; N. H. T. Nhat. Subgroups of the general linear group containing the elementary subgroup over a commutative ring extension of rank 2. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 31, Tome 455 (2017), pp. 209-225. http://geodesic.mathdoc.fr/item/ZNSL_2017_455_a15/

[1] Z. I. Borevich, N. A. Vavilov, “Subgroups of the general linear group over a commutative ring”, Dokl. Akad. N., 267:4 (1982), 777–778 | MR | Zbl

[2] L. N. Vaserstein, “Normal subgroups of the general linear groups over von Neumann regular rings”, J. Pur. Appl. Alg., 52 (1988), 187–195 | DOI | MR | Zbl

[3] L. N. Vaserstein, “On normal subgroups of $\mathrm{GL}_n$ over a ring”, Lect. Notes Math., 854, 1981, 456–465 | DOI | MR | Zbl

[4] S. Li, “Overgroups in $\mathrm{GL}(nr,F)$ of certain subgroups of $\mathrm{SL}(n,K)$”, J. Algebra, 125 (1989), 115–135 | MR

[5] A. A. Suslin, “The structure of the special linear group over rings of polynomials”, Izv. Akad. N. Ser. Mat., 41:2 (1977), 235–252 | MR | Zbl

[6] L. N. Vaserstein, A. A. Suslin, “Serre's problem on projective modules over polynomial rings, and algebraic K-theory”, Izv. Akad. N. Ser. Mat., 40:5 (1976), 993–1054 | MR | Zbl

[7] V. A. Stepanov, N. A. Vavilov, “Decomposition of transvections: a theme with variations”, K-Theory, 19 (2000), 109–153 | DOI | MR | Zbl

[8] N. A. Vavilov, V. A. Petrov, “On overgroups of $\mathrm{EO}(2l,R)$”, Zap. Nauch. Semin. POMI, 272, 2000, 68–85 | MR | Zbl

[9] N. A. Vavilov, V. A. Petrov, “On overgroups of $\mathrm{Ep}(2l,R)$”, Alg. Anal., 15:3 (2003), 72–114 | MR

[10] N. A. Vavilov, V. A. Petrov, “On overgroups of $\mathrm{EO}(n,R)$”, Alg. Anal., 19:2 (2007), 10–51 | MR

[11] J. Tits, “Systèmes générateurs de groupes de congruence”, C. R. Acad. Sci. Paris Ser A–B, 283 (1976), A693–A695 | MR

[12] N. H. T. Nhat, T. N. Hoi, “The normalizer of the elementary linear group of a module arising under extension of the base ring”, Zap. Nauch. Semin. POMI, 455, 2017, 122–129