Structure of small cancellation rings

Agatha Atkarskaya,Alexei Kanel-Belov,Eugene Plotkin,Eliyahu Rips

doi:10.5802/mrr.6

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

Highlights

The Small Cancellation Theory for groups is well known
Generated small cancellation groups turned out to be word hyperbolic
In the further argument we assume that τ 10

Summary

Introduction

The Small Cancellation Theory for groups is well known (see [13]). The similar theory exists for semigroups and monoids (see [10, 9, 22]). In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Generated small cancellation groups turned out to be word hyperbolic (when every relation needs at least 7 pieces). If we could generalize small cancellation to the ring-theoretic situation, it would provide examples to the yet undefined concept of a ring with a negative curvature Another source of potential examples are group algebras of hyperbolic groups. Following this reasoning, we introduce the three types of axioms for rings called Compatibility Axiom, Small Cancellation Axiom, and Isolation Axiom. A word s is called a small piece with respect to R (in generalized group sense, see [21, 13]) if there are relations of the form s.

Small cancellation axioms for the ring case

How we study the structure of small cancellation rings

Algorithmic properties

Examples