Group-like small cancellation theory for rings

A. Atkarskaya, A. Kanel-Belov, E. Plotkin, E. Rips

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
JournalInternational Journal of Algebra and Computation
StateAccepted/In press - 2023

Bibliographical note

Publisher Copyright:
© 2023 World Scientific Publishing Company.


  • 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


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