  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.
