Abstract
In this paper, we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a linear space and establish the corresponding structure theorems. We also provide a revision of a concept of Gröbner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.
| Original language | English |
|---|---|
| Pages (from-to) | 1269-1487 |
| Number of pages | 219 |
| Journal | International Journal of Algebra and Computation |
| Volume | 33 |
| Issue number | 7 |
| DOIs | |
| State | Published - 1 Nov 2023 |
Bibliographical note
Publisher Copyright:© World Scientific Publishing Company.
Funding
The research of the first, second and the third authors was supported by ISF grant 1994/20, 1207/12 and the Emmy Noether Research Institute for Mathematics. The research of the first and the fourth authors was also supported by ISF fellowship. The research of the second author was supported by the Russian Science Foundation, grant 22-11-00177, https://rscf.ru/project/22-11-00177/. We are very grateful to I. Kapovich, B. Kunyavskii and D. Osin for their invaluable cooperation.
| Funders | Funder number |
|---|---|
| Emmy Noether Research Institute for Mathematics | |
| Israel Science Foundation | 1207/12, 1994/20 |
| Russian Science Foundation | 22-11-00177 |
Keywords
- Dehn’s algorithm
- Gröbner basis
- Small cancellation ring
- defining relations in rings
- filtration
- greedy algorithm
- group algebra
- multi-turn
- small cancellation group
- tensor products
- turn