Construction of a quotient ring of z2 f in which a binomial 1 + w is invertible using small cancellation methods

A. Atkarskaya; A. Kanel-Belov; E. Plotkin; E. Rips

doi:10.1090/conm/726/14605

Construction of a quotient ring of z₂ f in which a binomial 1 + w is invertible using small cancellation methods

A. Atkarskaya, A. Kanel-Belov, E. Plotkin, E. Rips

Department of Mathematics

Hebrew University of Jerusalem

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Scopus citations

Abstract

We apply small cancellation methods originating from group theory to investigate the structure of a quotient ring Z₂ F/I where Z₂ F is the group algebra of the free group F over the field Z₂, and the ideal I is generated by a single trinomial 1 + v + vw, where v is a complicated word depending on w. In Z₂ F/I we have (1 + w)⁻¹ = v, so 1 + w becomes invertible. We construct an explicit linear basis of Z₂ F/I (thus showing that Z₂ F/I ≠ 0). This is the first step in constructing rings with exotic properties.

Original language	English
Title of host publication	Groups, Algebras and Identities
Editors	Eugene Plotkin
Publisher	American Mathematical Society
Pages	1-76
Number of pages	76
ISBN (Print)	9781470437138
DOIs	https://doi.org/10.1090/conm/726/14605
State	Published - 2019
Event	Research Workshop of the Israel Science Foundation on Groups, Algebras and Identities, 2016 - Jerusalem, Israel Duration: 20 Mar 2016 → 24 Mar 2016

Publication series

Name	Contemporary Mathematics
Volume	726
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Conference

Conference	Research Workshop of the Israel Science Foundation on Groups, Algebras and Identities, 2016
Country/Territory	Israel
City	Jerusalem
Period	20/03/16 → 24/03/16

Bibliographical note

Publisher Copyright:
© 2019 A. Atkarskaya, A. Kanel-Belov, E. Plotkin, E. Rips.

Access to Document

10.1090/conm/726/14605

Cite this

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title = "Construction of a quotient ring of z2 f in which a binomial 1 + w is invertible using small cancellation methods",

abstract = "We apply small cancellation methods originating from group theory to investigate the structure of a quotient ring Z2 F/I where Z2 F is the group algebra of the free group F over the field Z2, and the ideal I is generated by a single trinomial 1 + v + vw, where v is a complicated word depending on w. In Z2 F/I we have (1 + w)−1 = v, so 1 + w becomes invertible. We construct an explicit linear basis of Z2 F/I (thus showing that Z2 F/I ≠ 0). This is the first step in constructing rings with exotic properties.",

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Atkarskaya, A, Kanel-Belov, A, Plotkin, E & Rips, E 2019, Construction of a quotient ring of z₂ f in which a binomial 1 + w is invertible using small cancellation methods. in E Plotkin (ed.), Groups, Algebras and Identities. Contemporary Mathematics, vol. 726, American Mathematical Society, pp. 1-76, Research Workshop of the Israel Science Foundation on Groups, Algebras and Identities, 2016, Jerusalem, Israel, 20/03/16. https://doi.org/10.1090/conm/726/14605

Construction of a quotient ring of z₂ f in which a binomial 1 + w is invertible using small cancellation methods. / Atkarskaya, A.; Kanel-Belov, A.; Plotkin, E. et al.
Groups, Algebras and Identities. ed. / Eugene Plotkin. American Mathematical Society, 2019. p. 1-76 (Contemporary Mathematics; Vol. 726).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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