Construction of a quotient ring of $\mathbb{Z}_2\mathcal{F}$ in which a binomial $1 + w$ is invertible using small cancellation methods
| Autor: | Atkarskaya, A., Kanel-Belov, A., Plotkin, E., Rips, E. |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | We apply small cancellation methods originating from group theory to investigate the structure of a quotient ring $\mathbb{Z}_2\mathcal{F} / \mathcal{I}$, where $\mathbb{Z}_2\mathcal{F}$ is the group algebra of the free group $\mathcal{F}$ over the field $\mathbb{Z}_2$, and the ideal $\mathcal{I}$ is generated by a single trinomial $1 + v + vw$, where $v$ is a complicated word depending on $w$. In $\mathbb{Z}_2\mathcal{F} / \mathcal{I}$ we have $(1 + w)^{-1} = v$, so $1 + w$ becomes invertible. We construct an explicit linear basis of $\mathbb{Z}_2\mathcal{F} / \mathcal{I}$ (thus showing that $\mathbb{Z}_2\mathcal{F} / \mathcal{I}\neq 0$). This is the first step in constructing rings with exotic properties. Comment: To be published in Contemporary Mathematics, Israel Mathematical Conference Proceedings (IMCP), 2019 Reference to a grant is added |
| Databáze: | arXiv |
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