Stratifying Systems via Nested Family of Torsion Pairs

In this paper, we introduce the concept of a nested family of torsion pairs and will prove that this concept is strongly related to the existence of stratifying systems. Specifically, every stratifying system induces a nested family of torsion pairs. Moreover, every stratifying system can be obtained as a quotient or submodule of a module that admits a certain direct sum decomposition with respect to the nested family of torsion pairs. Additionally, we present a stratifying system of infinite size that cannot be indexed by $\mathbb{N}$.