Structure of small cancellation rings

Abstract

The theory of small cancellation groups is well known. In this paper we study the notion of the Group-like Small Cancellation Ring. 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 and 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 Gröbner 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.