Structure of small cancellation rings
Mathematics Research Reports, Volume 2 (2021) , pp. 1-14.

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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/mrr.6
Classification: 20F67,  16S15,  16Z05
Keywords: small cancellation ring, turn, multi-turn, defining relations in rings, small cancellation group, group algebra, filtration, tensor products, Dehn’s algorithm, greedy algorithm, Gröbner basis
@article{MRR_2021__2__1_0,
author = {Agatha Atkarskaya and Alexei Kanel-Belov and Eugene Plotkin and Eliyahu Rips},
title = {Structure of small cancellation rings},
journal = {Mathematics Research Reports},
pages = {1--14},
publisher = {MathOA foundation},
volume = {2},
year = {2021},
doi = {10.5802/mrr.6},
language = {en},
url = {https://mrr.centre-mersenne.org/articles/10.5802/mrr.6/}
}
Agatha Atkarskaya; Alexei Kanel-Belov; Eugene Plotkin; Eliyahu Rips. Structure of small cancellation rings. Mathematics Research Reports, Volume 2 (2021) , pp. 1-14. doi : 10.5802/mrr.6. https://mrr.centre-mersenne.org/articles/10.5802/mrr.6/

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