Group-like small cancellation theory for rings

TL;DR

This paper develops a small cancellation theory for associative algebras with invertible basis elements, introducing axioms for relations, proving non-triviality, and establishing structural properties including a basis and filtration.

Contribution

It introduces a novel small cancellation framework for rings, extending group theory concepts to associative algebras, with new structure theorems and computational tools.

Findings

The ring is proven to be non-trivial.

A basis for the ring as a vector space is constructed.

A greedy algorithm for the Ideal Membership Problem is established.

Abstract

In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. We also provide a revision of a concept of Gr\"{o}bner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.