Decomposition of torsion pairs on module categories

In this article, we generalize the concept of torsion pair and study its structure. As a trial of obtaining all torsion pairs, we decompose torsion pairs by projective modules and injective modules. Then we calculate torsion pairs on the algebra KAnKAn and tube categories. At last we study the structure of torsion pairs on the module categories of finite-dimensional hereditary algebras.