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

TL;DR

This paper uses small cancellation techniques from group theory to construct a quotient ring of a group algebra over ield where a specific binomial becomes invertible, providing an explicit basis and advancing the understanding of rings with unusual properties.

Contribution

It introduces a novel application of small cancellation methods to construct a quotient ring where a binomial is invertible, with an explicit basis for the structure.

Findings

Explicit basis for the quotient ring constructed.

Proof that the quotient ring is non-zero.

Demonstration of invertibility of a binomial in the quotient.

Abstract

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.