F-Noetherian and Weakly Exact Torsion Theories

It is well-known (see [13]) that a hereditary torsion theory tau for the category R-mod is noetherian if and only if the class of all tau-torsionfree tau-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 tau-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.