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.
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Mots clés : 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 = {Atkarskaya, Agatha and Kanel-Belov, Alexei and Plotkin, Eugene and Rips, Eliyahu}, 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 = {http://www.numdam.org/articles/10.5802/mrr.6/} }
TY - JOUR AU - Atkarskaya, Agatha AU - Kanel-Belov, Alexei AU - Plotkin, Eugene AU - Rips, Eliyahu TI - Structure of small cancellation rings JO - Mathematics Research Reports PY - 2021 SP - 1 EP - 14 VL - 2 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/mrr.6/ DO - 10.5802/mrr.6 LA - en ID - MRR_2021__2__1_0 ER -
%0 Journal Article %A Atkarskaya, Agatha %A Kanel-Belov, Alexei %A Plotkin, Eugene %A Rips, Eliyahu %T Structure of small cancellation rings %J Mathematics Research Reports %D 2021 %P 1-14 %V 2 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/mrr.6/ %R 10.5802/mrr.6 %G en %F MRR_2021__2__1_0
Atkarskaya, Agatha; Kanel-Belov, Alexei; Plotkin, Eugene; Rips, Eliyahu. Structure of small cancellation rings. Mathematics Research Reports, Tome 2 (2021), pp. 1-14. doi : 10.5802/mrr.6. http://www.numdam.org/articles/10.5802/mrr.6/
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