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Ore Localisation for differential graded rings; Towards Goldie's theorem for differential graded algebras

Alexander Zimmermann

Abstract

We study Ore localisation of differential graded algebras. Further we define dg-prime rings, dg-semiprime rings, and study the dg-nil radical of dg-rings. Then, we define dg-essential submodules, dg-uniform dimension, and apply all this to a dg-version of Goldie's theorem on prime dg-rings.

Ore Localisation for differential graded rings; Towards Goldie's theorem for differential graded algebras

Abstract

We study Ore localisation of differential graded algebras. Further we define dg-prime rings, dg-semiprime rings, and study the dg-nil radical of dg-rings. Then, we define dg-essential submodules, dg-uniform dimension, and apply all this to a dg-version of Goldie's theorem on prime dg-rings.
Paper Structure (16 sections, 35 theorems, 71 equations)

This paper contains 16 sections, 35 theorems, 71 equations.

Table of Contents

  1. The classical situation: Goldie's theorem and its graded version
  2. The classical ungraded case
  3. The graded situation
  4. Ore localisation and differential graded rings
  5. Classical Ore localisation revisited
  6. Localisation of dg-algebras making homology classes invertible
  7. Extending the differential to the Ore localisation
  8. Localisation of dg-rings at homogeneous elements
  9. Application: Goldie's theorem
  10. Comparing homology localisation with Ore localisation
  11. Semiprime differential graded rings
  12. dg-prime, dg-semiprime
  13. dg-essential dg-submodules
  14. dg-uniform dimension, the dg-singular ideal
  15. dg-Goldie rings; left and right dg-annihilators
  16. ...and 1 more sections

Key Result

Theorem 1.1

(Goldie) The following are equivalent. Further, in the last item $R$ is prime if and only if $Q$ is simple.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Corollary 2.8
  • ...and 54 more