Melkersson conditions with respect to a prime ideal

TL;DR

This paper explores the structure of prime ideals satisfying Melkersson conditions in module categories over various rings, providing classifications for specific low-dimensional cases and analyzing extension modules.

Contribution

It introduces methods to determine prime ideals meeting Melkersson conditions and classifies their structure over rings of dimensions 0, 1, and 2.

Findings

Characterization of prime ideals satisfying Melkersson conditions

Classification over 0-dimensional rings

Analysis of prime ideals over 1- and 2-dimensional local rings

Abstract

Aghapournahr and Melkersson introduced the notion of Melkersson condition on a Serre subcategory of the module category over a commutative noetherian ring. This paper investigates the structure of set of prime ideals satisfying a Melkersson condition on a Serre subcategory. We try to calculate members of a set of these prime ideals for a subcategory consisting of extension modules in two given Serre subcategories of the module category. Meanwhile, we classify the structure of set of prime ideals satisfying a Melkersson condition over a 0-dimensional ring, a 1-dimensional local ring, and a 2-dimensional local domain.