The Characterization Of Weak Finiteness Under The Hereditary Torsion Theory

Hereditary torsion theory has received a good deal of attention continuously in a number of literature in recent decade. In this paper, first, we apply the notion of hereditary torsion theory and its corresponding results to characterize weak finitely under the hereditary torsion theory, introduce the concept ofτ-torsionfree injective modules, and then generalize 5 Lemma, Snake Lemma,9 Lemma and Schanuel Lemma. We further introduceτ-injective module, and prove that not all modules haveτ-nature extention andτ-torsion free module haveτ-injective envelop. Second, we give the equivalence characterization ofτ-noetherian rings andτ-coherent rings. For a ring R, the following statements are equivalent:(ⅰ) R is aτ-coherent ring; (ⅱ) everyτ-finitely generated module of R isτ-finitely pre-sented; (ⅲ) everyτ-finitely generated submodule ofτ-finitely presented R-module isτ-finitely presented. Then we obtain the following equivelent conditions:(ⅰ) R, isτ-noetherian ring; (ⅱ) every prime ideal of R isτ-finitely generated; (ⅲ) anyτ-finitely generated R-module isτ-noetherian module; (ⅳ) anyτ-finitely gener-ated R-module isτ-finitely presented (ⅴ) the oplus ofτ-injective R-modules isτ-injective R-module.