Normal Subgroups of the General Linear Groups Over Von Neumann Regular Rings

Vaserstein, L. N.

doi:10.2307/2046154

Cited by 5 publications

(5 citation statements)

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“…Similar results for other classes of rings (including the class of commutative rings) were obtained in [3], [11], [4], [8], [9] and [10].…”

Section: Introductionsupporting

confidence: 80%

On general linear groups over exchange rings

Preusser

2020

Linear and Multilinear Algebra

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Let R be an exchange ring. We prove that the relative elementary subgroups E n (R, I) are normal in the general linear group GL n (R) if n ≥ 1 and that the standard commutator formula E n (R, I) = [E n (R), E n (R, I)] = [E n (R), C n (R, I)] holds if n ≥ 3. Moreover, we classify the subgroups of GL n (R) that are normalised by the elementary subgroup E n (R) in the case n ≥ 3.2010 Mathematics Subject Classification. 15A24, 20G35.

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“…Similar results for other classes of rings (including the class of commutative rings) were obtained in [3], [11], [4], [8], [9] and [10].…”

Section: Introductionsupporting

confidence: 80%

On general linear groups over exchange rings

Preusser

2020

Linear and Multilinear Algebra

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“…), for all n ^ 3. (He has extended this result [15], [16], [17] to subgroups of Gl n (S), for particular classes of non-commutative rings 5, where n ^ 3.) For Vaserstein's result (or an even weaker version of his result) to carry over to the case where n = 2 it appears that R has to contain "sufficiently many units".…”

mentioning

confidence: 87%

Non-Standard, Normal Subgroups and Non-Normal, Standard Subgroups of the Modular Group

Mason¹

1989

Can. math. bull.

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Let R be a commutative ring with identity. A subgroup S of GLn(R), where n ≥ 2, is said to be standard if and only if S contains all the q-elementary matrices and all conjugates of those matrices by products of elementary matrices, where q is the ideal in R generated by Xij,xii — Xjj(i ≠ j), for all (xij) ∊ S. It is known that, when n ≧ 3, the standard subgroups of GLn(R) are precisely those normalized by the elementary matrices. To demonstrate how completely this result can break down for n = 2 we prove that GL2(Z), where Z is the ring of rational integers, has uncountably many non-normal, standard subgroups and uncountably many non-standard, normal subgroups.

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“…It turns out [19,30] that von Neumann regular rings are stable. Indeed, let R be a von Neumann regular ring and g ∈ GL (n, R) .…”

Section: Definition 11mentioning

confidence: 99%

Stability of rings

Petechuk

2010

Preprint

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The conditions for stability of the elements of linear groups over the associative rings with identity and their connection with the stability of rings are analyzed in the article. The stability of rings which are commutative, satisfy the conditions of stability of rank ≥ 2, von Neumann regular, integer-algebraic, nearly local rings introduced by the author, is examined. The most important classical results of H. Bass, L.

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Normal Subgroups of the General Linear Groups Over Von Neumann Regular Rings

Cited by 5 publications

References 6 publications

On general linear groups over exchange rings

On general linear groups over exchange rings

Non-Standard, Normal Subgroups and Non-Normal, Standard Subgroups of the Modular Group

Stability of rings

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