Commutators of relative and unrelative elementary groups, revisited

Standard

Commutators of relative and unrelative elementary groups, revisited. / Вавилов, Николай Александрович; Zhang, Zuhong.

In: Journal of Mathematical Sciences (United States), Vol. 251, No. 3, 01.12.2020, p. 339-348.

Research output: Contribution to journal › Review article › peer-review

BibTeX

@article{54f94ca6806748abab8f318126850a08,

title = "Commutators of relative and unrelative elementary groups, revisited",

abstract = "Let R be any associative ring with 1, n > 3, and let A,B be two-sided ideals of R. In the present paper we show that the mixed commutator subgroup [E(n,R,A),E(n,R,B)] is generated as a group by the elements of the two following forms: 1) z_ij (ab,c) and z_ij (ba,c), 2) [t_ij (a), t_ji(b)], where 1<= i 6/= j <=n, a ∈ A, b ∈ B, c ∈ R. Moreover, for the second type of generators, it suffices to fix one pair of indices (i, j). This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies that for all associative rings one has the equality [E(n,R,A),E(n,R,B)] = [E(n,A),E(n,B)] and many further corollaries can be derived for rings subject to commutativity conditions. ",

keywords = "General linear group, elementary subgroup, Congruence subgroups, standard commutator formulae, unrelativised commutator formula, elementary generators",

author = "Вавилов, {Николай Александрович} and Zuhong Zhang",

year = "2020",

month = dec,

day = "1",

doi = "10.1007/s10958-020-05094-4",

language = "English",

volume = "251",

pages = "339--348",

journal = "Journal of Mathematical Sciences",

issn = "1072-3374",

publisher = "Springer Nature",

number = "3",

}

RIS

TY - JOUR

T1 - Commutators of relative and unrelative elementary groups, revisited

AU - Вавилов, Николай Александрович

AU - Zhang, Zuhong

PY - 2020/12/1

Y1 - 2020/12/1

N2 - Let R be any associative ring with 1, n > 3, and let A,B be two-sided ideals of R. In the present paper we show that the mixed commutator subgroup [E(n,R,A),E(n,R,B)] is generated as a group by the elements of the two following forms: 1) z_ij (ab,c) and z_ij (ba,c), 2) [t_ij (a), t_ji(b)], where 1<= i 6/= j <=n, a ∈ A, b ∈ B, c ∈ R. Moreover, for the second type of generators, it suffices to fix one pair of indices (i, j). This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies that for all associative rings one has the equality [E(n,R,A),E(n,R,B)] = [E(n,A),E(n,B)] and many further corollaries can be derived for rings subject to commutativity conditions.

AB - Let R be any associative ring with 1, n > 3, and let A,B be two-sided ideals of R. In the present paper we show that the mixed commutator subgroup [E(n,R,A),E(n,R,B)] is generated as a group by the elements of the two following forms: 1) z_ij (ab,c) and z_ij (ba,c), 2) [t_ij (a), t_ji(b)], where 1<= i 6/= j <=n, a ∈ A, b ∈ B, c ∈ R. Moreover, for the second type of generators, it suffices to fix one pair of indices (i, j). This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies that for all associative rings one has the equality [E(n,R,A),E(n,R,B)] = [E(n,A),E(n,B)] and many further corollaries can be derived for rings subject to commutativity conditions.

KW - General linear group

KW - elementary subgroup

KW - Congruence subgroups

KW - standard commutator formulae

KW - unrelativised commutator formula

KW - elementary generators

UR - http://www.scopus.com/inward/record.url?scp=85094671788&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/04395f40-3657-37c2-b535-80692872b652/

U2 - 10.1007/s10958-020-05094-4

DO - 10.1007/s10958-020-05094-4

M3 - Review article

VL - 251

SP - 339

EP - 348

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 3

ER -

ID: 61526731