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Group-like small cancellation theory for rings

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

Abstract

In the present 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 vector 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.

Group-like small cancellation theory for rings

Abstract

In the present 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 vector 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.

Paper Structure

This paper contains 33 sections, 82 theorems, 518 equations.

Table of Contents

  1. Introduction
  2. Motivation, objectives, results
  3. Overview of the work
  4. Small cancellation for groups
  5. Main definitions and examples for the ring case
  6. Structure of small cancellation algebras
  7. A posteriori insight. Gröbner basis
  8. Route map of the paper
  9. Group-like small cancellation axioms
  10. Basic definitions
  11. The description of the ideal $\mathcal{I}$ as a linear subspace of $\mathit{k}\mathcal{F}$
  12. How multi-turns influence maximal occurrences
  13. Virtual members of the chart
  14. Images of maximal occurrences
  15. Minimal coverings of a monomial
  16. ...and 18 more sections

Key Result

Theorem 1

$\mathrm{F}_n (\mathit{k}\mathcal{F}) \cap \mathcal{I}$ is linearly spanned by all the polynomials of the form $L^{(i)} \cdot p\cdot R^{(i)}$, $i = 1,\ldots , m$ for all monomials $U \in \mathrm{F}_n (\mathit{k}\mathcal{F})$ and polynomials $p \in \mathcal{R}$ as above, $n \geqslant 0$.

Theorems & Definitions (213)

  • Remark 1.1
  • Theorem 1
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.1
  • ...and 203 more