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The Artin--Rees lemma and size of spaces over nonassociative complete filtered rings

Jayanta Manoharmayum

Abstract

This paper studies nonassociative filtered rings using associated gradations. We show that a complete filtered ring $R$ with affine associated graded ring generated in degree $1$ is a local ring, and prove that the Artin--Rees lemma holds for $R$. Assuming finiteness of the residue field, we derive asymptotics for abelian groups with an operation of $R$, and identify classes of torsion elements for spaces over a central extension of $R$.

The Artin--Rees lemma and size of spaces over nonassociative complete filtered rings

Abstract

This paper studies nonassociative filtered rings using associated gradations. We show that a complete filtered ring with affine associated graded ring generated in degree is a local ring, and prove that the Artin--Rees lemma holds for . Assuming finiteness of the residue field, we derive asymptotics for abelian groups with an operation of , and identify classes of torsion elements for spaces over a central extension of .
Paper Structure (19 sections, 27 theorems, 76 equations)

This paper contains 19 sections, 27 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Finiteness conditions for spaces over a ring
  3. Generalities
  4. Hilbert's basis theorem
  5. Filtered rings and spaces
  6. Filtered abelian groups
  7. Filtered rings and filtered spaces
  8. The Artin--Rees lemma for complete local-filtered rings
  9. The Rees ring
  10. Complete local-filtered rings
  11. The Artin-Rees lemma
  12. Dimensions and sizes of permissible spaces
  13. Independence of permissible filtrations
  14. Size of permissible spaces
  15. Torsion elements in permissible spaces
  16. ...and 4 more sections

Key Result

Proposition 2.4

Let $R$ be a ring and let $M$ be an $R$-space. Then the following statements hold.

Theorems & Definitions (69)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6: Hilbert's basis theorem
  • proof : Proof of Proposition \ref{['P:hbt']}\ref{['hbt1']}
  • proof : Proof of Proposition \ref{['P:hbt']}\ref{['hbt2']}
  • Lemma 3.1
  • ...and 59 more