An example of Melkersson subcategory which is not closed under injective hulls

TL;DR

This paper demonstrates that not all Melkersson subcategories are closed under injective hulls, answering an open question in the theory of Serre subcategories negatively.

Contribution

It provides a counterexample showing that the converse of a known implication does not hold in the context of Melkersson subcategories.

Findings

Counterexample disproves the converse implication

Clarifies the relationship between Melkersson subcategories and injective hulls

Advances understanding of Serre subcategory properties

Abstract

The Melkersson subcategory is a special Serre subcategory. It was proved that a Serre subcategory which is closed under injective hulls is a Melkersson subcategory. However, it has been an open question whether the contrary implication holds. In this paper, we shall show that this question has a negative answer in general.