## Abstract

We apply small cancellation methods originating from group theory to investigate the structure of a quotient ring Z_{2} F/I where Z_{2} F is the group algebra of the free group F over the field Z_{2}, and the ideal I is generated by a single trinomial 1 + v + vw, where v is a complicated word depending on w. In Z_{2} F/I we have (1 + w)^{−1} = v, so 1 + w becomes invertible. We construct an explicit linear basis of Z_{2} F/I (thus showing that Z_{2} F/I ≠ 0). This is the first step in constructing rings with exotic properties.

Original language | English |
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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 | |

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 |
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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 |
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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.

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