ℱ-Noetherian and Weakly Exact Torsion Theories

Abstract

It is well-known (see [13]) that a hereditary torsion theory τ for the category R-mod is noetherian if and only if the class of all τ-torsionfree τ-injective modules is closed under arbitrary direct sums. So, it is natural to investigate the hereditary torsion theories having the property that the class of all τ-torsionfree injective modules is closed under arbitrary direct sums, which are called ℱ-noetherian. These torsion theories have been studied by Teply in [16]. In the second part of this note we shall study the weakly exact hereditary torsion theories, which generalize the exact one's.

Key Words:

• Cover class
• Exact torsion theory
• Hereditary torsion theory
• Left noetherian ring
• Noetherian torsion theory
• Precover class

2010 Mathematics Subject Classification:

• 16S90
• 16D80
• 16P40
• 16D90

Notes

Communicated by T. Albu.