Locally prime modules

TL;DR

This paper introduces and studies locally prime modules over commutative rings, establishing dualities and equivalences that connect local primality concepts with classical global prime modules, using category-theoretic methods.

Contribution

It defines $I$-prime and $(I,J)$-prime modules, extends duality theories to these modules, and links local primality to classical global primality.

Findings

Established Greenlees-May Duality for locally prime modules.

Proved Matlis-Greenlees-May Equivalence in this context.

Connected local and global primality concepts through module theory.

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.