On the closedness of taking injective hulls of several Serre subcategories

ABSTRACT

Belshoff and Xu showed that every Matlis reflexive module has a Matlis reflexive injective hull if and only if R is complete and has dimension less than or equal to 1. In this paper, we give a characterization of the closedness of taking injective hulls for a Serre subcategory consisting of Minimax modules. In addition, the closedness of taking injective hulls for a Serre subcategory consisting of extension modules of finitely generated modules by modules with finite support is characterized by the number of prime ideals. Our results provide a negative answer to Aghapournahr and Melkersson’s question concerning Melkersson subcategories.

KEYWORDS:

• Matlis reflexive module
• Melkersson subcategory
• minimax module
• Serre subcategory

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 13C60
• 13D45

Acknowledgments

The author expresses gratitude to the referees for their kind comments and valuable suggestions. Thanks to referee’s advice, the author could drop the assumption that a ring is local in Theorem 3.5.