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Locally prime modules

Sholastica Luambano, David Ssevviiri

Abstract

For a commutative unital ring $R$ with fixed ideals $I$ and $J$, we introduce and study $I$-prime $R$-modules and $(I, J)$-prime $R$-modules together with their duals $I$-coprime $R$-modules and $(I,J)$-coprime $R$-modules respectively. We employ category-theoretic techniques to reveal their structural properties. Our main results are versions of the Greenlees-May Duality and the Matlis-Greenlees-May Equivalence to the setting of these prime and coprime modules. This generalizes work on $I$-reduced modules and $I$-coreduced modules. We demonstrate that these ``locally prime" modules serve as a tool for studying the classical ``globally prime" modules, creating a bridge between local and global primality.

Locally prime modules

Abstract

For a commutative unital ring with fixed ideals and , we introduce and study -prime -modules and -prime -modules together with their duals -coprime -modules and -coprime -modules respectively. We employ category-theoretic techniques to reveal their structural properties. Our main results are versions of the Greenlees-May Duality and the Matlis-Greenlees-May Equivalence to the setting of these prime and coprime modules. This generalizes work on -reduced modules and -coreduced modules. We demonstrate that these ``locally prime" modules serve as a tool for studying the classical ``globally prime" modules, creating a bridge between local and global primality.
Paper Structure (16 sections, 24 theorems, 7 equations, 3 tables)

This paper contains 16 sections, 24 theorems, 7 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Globally prime modules
  3. Locally prime modules
  4. Torsion and Completion
  5. Summary of results
  6. Rationale for the GM Duality in the context of modules
  7. Structure of the paper
  8. $I$-prime modules $\&$ $(I, J)$-prime modules
  9. $I$-prime $R$-modules
  10. $(I,J)$-prime modules
  11. The dual notions of locally prime modules
  12. Globally coprime modules
  13. Locally coprime modules
  14. GM Duality and MGM Equivalence
  15. More functorial properties
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $I$ be a fixed ideal of $R$ and $M$ be an $R$-module. The following statements are equivalent:

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Proposition 2.1
  • Corollary 2.1
  • Corollary 2.2
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.1
  • ...and 20 more