Structure of small cancellation rings (Q2147360)

The small cancellation theory is widely studied for groups. Monoids and semigroups also share a theory in similar. However, developing such a theory for systems with two operations is extremely challenging. In the current study, the authors offer a minimal cancellation theory for invertible element-based associative algebras. They axiomatically describe a ring that can be appropriately referred to as a ring with small cancellation properties and conditions, namely satisfies three types of axioms for rings called Compatibility Axiom, Small Cancellation Axiom, and Isolation Axiom. These small cancellation rings are non-trivial, have a global filtration that agrees with the relationships, have algorithmic properties similar to those that apply to groups with small cancellation, and also have a Gröbner basis with respect to some sophisticated linear order on monomials, according to the authors' findings. In addition to building an explicit basis for the ring as a linear space, the corresponding structure theorems are also demonstrated.