Construction of a quotient ring of ${\mathbb{Z}}_2\mathcal{F}$ in which a binomial $1+w$ is invertible using small cancellation methods

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 ${\mathbb{Z}}_2\mathcal{F}\mathrm{/}\mathcal{I}$ we have $(1 + w)^{-1} = v$,so$1 + w$ becomes invertible. We construct an explicit linear basis of ${\mathbb{Z}}_2\mathcal{F}\mathrm{/}\mathcal{I}$ (thus showing that ${\mathbb{Z}}_2\mathcal{F}\mathrm{/}\mathcal{I}\mathrm{\ne }0$). This is the first step in constructing rings with exotic properties.