Multiple commutator formulas

Roozbeh Hazrat; Zuhong Zhang

doi:10.1007/s11856-012-0135-8

Multiple commutator formulas

Roozbeh Hazrat^*, Zuhong Zhang

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

16 Citations (Scopus)

Abstract

Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I _i, i = 0,...,m, be two-sided ideals of A, GL_n(A, I _i) be the principal congruence subgroup of level I _i in GL_n(A) and E _n(A, I _i) be the relative elementary subgroup of level I _i. We prove the multiple commutator formula [E_n(A, I₀), GL_n(A, I₁), GL_n(A, I₂), ... , GL_n(A, I_m)] = [E_n(A, I₀), E_n(A, I₁), E_n(A, I₂), ... , E_n(A, I_m)], which is a broad generalization of the standard commutator formulas. This result contains all the published results on commutator formulas over commutative rings and answers a problem posed by A. Stepanov and N. Vavilov.

Original language	English
Pages (from-to)	481-505
Number of pages	25
Journal	Israel Journal of Mathematics
Volume	195
Issue number	1
DOIs	https://doi.org/10.1007/s11856-012-0135-8
Publication status	Published - Jun 2013

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Hazrat, R., & Zhang, Z. (2013). Multiple commutator formulas. Israel Journal of Mathematics, 195(1), 481-505. https://doi.org/10.1007/s11856-012-0135-8

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abstract = "Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I i, i = 0,...,m, be two-sided ideals of A, GLn(A, I i) be the principal congruence subgroup of level I i in GLn(A) and E n(A, I i) be the relative elementary subgroup of level I i. We prove the multiple commutator formula [En(A, I0), GLn(A, I1), GLn(A, I2), ... , GLn(A, Im)] = [En(A, I0), En(A, I1), En(A, I2), ... , En(A, Im)], which is a broad generalization of the standard commutator formulas. This result contains all the published results on commutator formulas over commutative rings and answers a problem posed by A. Stepanov and N. Vavilov.",

author = "Roozbeh Hazrat and Zuhong Zhang",

year = "2013",

month = jun,

doi = "10.1007/s11856-012-0135-8",

language = "English",

volume = "195",

pages = "481--505",

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AU - Hazrat, Roozbeh

AU - Zhang, Zuhong

PY - 2013/6

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N2 - Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I i, i = 0,...,m, be two-sided ideals of A, GLn(A, I i) be the principal congruence subgroup of level I i in GLn(A) and E n(A, I i) be the relative elementary subgroup of level I i. We prove the multiple commutator formula [En(A, I0), GLn(A, I1), GLn(A, I2), ... , GLn(A, Im)] = [En(A, I0), En(A, I1), En(A, I2), ... , En(A, Im)], which is a broad generalization of the standard commutator formulas. This result contains all the published results on commutator formulas over commutative rings and answers a problem posed by A. Stepanov and N. Vavilov.

AB - Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I i, i = 0,...,m, be two-sided ideals of A, GLn(A, I i) be the principal congruence subgroup of level I i in GLn(A) and E n(A, I i) be the relative elementary subgroup of level I i. We prove the multiple commutator formula [En(A, I0), GLn(A, I1), GLn(A, I2), ... , GLn(A, Im)] = [En(A, I0), En(A, I1), En(A, I2), ... , En(A, Im)], which is a broad generalization of the standard commutator formulas. This result contains all the published results on commutator formulas over commutative rings and answers a problem posed by A. Stepanov and N. Vavilov.

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DO - 10.1007/s11856-012-0135-8

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SN - 0021-2172

VL - 195

SP - 481

EP - 505

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

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Multiple commutator formulas

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