Small cancelation rings

TL;DR

This paper introduces Group-like Small Cancellation Rings, defining them axiomatically and proving they are non-trivial, with a structured basis, filtration, and algorithmic properties, aiming to develop a new theory analogous to small cancellation groups.

Contribution

The paper defines a new class of rings called Group-like Small Cancellation Rings, establishing their structure, basis, and properties, and proposing their use in advanced ring theory research.

Findings

The ring is non-trivial.

It has a well-defined basis and filtration.

It possesses a Gr"obner basis and greedy algorithm.

Abstract

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. 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. It turns out that the defined ring possesses a kind of Gr\"obner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory. This is a short version of paper arXiv:2010.02836