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