We continue studying ICE-closed subcategories of module categories (subcategories closed under taking Images, Cokernels and Extensions) from my previous paper, where the hereditary case was studied. We succeed in generalizing this for any algebras, introducing wide $\tau$-tilting modules, a $\tau$-tilting object in some functorially finite wide subcategory.
To achieve this, we study ICE-closed subcategories via the hearts of intervals in the lattice of torsion classes (we borrow the word heart from the Tattar’s paper, and is used in DIRRT, Asai-Pfeifer, and so on). We prove that every ICE-closed subcats are realized as hearts of some intervals, and characterize such intervals in a purely lattice-theoretic way. Using this, we show that ICE-closed subcategories are precisely torsion classes of some wide subcategories. This enables us to use tau-tilting theory to classify ICE-closed subcategories. Moreover, we discuss how to compute wide tau-tilting modules from the support tau-tilting poset for the tau-tilting finite case.