In the paper, we describe when τ-weak projective modules are closed under taking homomorphic images, which is dual to τ-injective module; we also describe a distinguish theorem on τ-weak projective cover.

In the paper,we prove Schanuel s lemma of relatively herediary torsion theory by τ-weak projective module and prove the dual of Schanuel s lemma by τ-injective module.

In the paper, we describe when τ-weak projective modules are closed under taking homomorphic images, which is dual to τ-injective module; we also describe a distinguish theorem on τ-weak projective cover.

Introduce　the　corresponding　concepts　of　cover　and　hull　through　Q　gets　the　properties　of τ-injective,τ-projective,τ-divisible　and　τ-codivisible,discuss　the　properties　of　τ-torsion　injective　hull, τ-torsion　projective　cover,τ-torsionfree　divisible　hull　and　τ-torsionfree　codivisible　cover, τ-injective　hull　E_τ(M)　is　characterized　by　using　them.

The existence of τ-projective test modules is investigated,and some properties of τ-projective modules are given by using dimensions.

τ-Projective test modules and τ-projective dimensions

We obtainthat a preprojective component Γ of Γ_A is a τ-preprojective component of Γ_A if Γcontains no oriented cycles, and that Γ is a preprojective componeat of Γ_A if and onlyif Γ is a τ-preprojective component of Γ_A for a quasitilted algebra A.