Small cancellation rings

All the rings we deal with are algebras over a field with a basis of invertible elements. The small cancellation theory for groups (along with its generalizations) is a well-known and quite useful part of the combinatorial theory of groups. When a group is given by generators and defining relations, the small cancellation condition means that in a certain sense there is a weak interaction between the defining relations. In order to define an analogous notion for rings, we formulate an appropriate Small Cancellation axiom. However, we need to add what we call the Isolation axiom. Then we develop the structure theory for rings satisfying these axioms. We construct a filtration on a small cancellation ring and find a linear basis of it. In particular, we show that the small cancellation ring is non-trivial. We anticipate that many of the properties of small cancellation groups or of hyperbolic groups have appropriate analogs in the case of small cancellation rings. On the other hand, one can consider rings with small cancellation as the first step towards a more general theory of rings with iterated small cancellation yet to be constructed, that could yield examples of rings with exotic properties.