In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval \([\mathcal {U},\mathcal {T}]\) in the lattice of torsion classes, we associate a subcategory \(\mathcal {T} \cap \mathcal {U}^\perp \) called the heart. We show that every ICE-closed subcategory can be realized as a heart of some interval of torsion classes, and give a lattice-theoretic characterization of intervals whose hearts are ICE-closed. In particular, we prove that ICE-closed subcategories are precisely torsion classes in some wide subcategories. For an artin algebra, we introduce the notion of wide \(\tau \)-tilting modules as a generalization of support \(\tau \)-tilting modules. Then we establish a bijection between wide \(\tau \)-tilting modules and doubly functorially finite ICE-closed subcategories, which extends Adachi–Iyama–Reiten’s bijection on torsion classes. For the hereditary case, we discuss the Hasse quiver of the poset of ICE-closed subcategories by introducing a mutation of rigid modules.

在本文中，我们使用扭转类研究阿贝尔长度类别的 ICE 封闭（= Image-Cokernel-Extension-closed）子类别。对于扭转类格中的每个区间\([\mathcal {U},\mathcal {T}]\)，我们关联一个子类别\(\mathcal {T} \cap \mathcal {U}^\perp \)称为心。我们表明，每个 ICE 闭合的子类别都可以实现为某个扭转类区间的心脏，并给出了心脏是 ICE 闭合的区间的格子理论表征。特别是，我们证明了 ICE 封闭的子类别正是一些宽子类别中的扭转类。对于艺术代数，我们引入了宽\(\tau \)倾斜模块的概念作为支持\(\tau \)- 倾斜模块。然后，我们在宽\(\tau \)倾斜模块和双函数有限 ICE 封闭子类别之间建立双射，它扩展了 Adachi-Iyama-Reiten 在扭转类上的双射。对于遗传情况，我们通过引入刚性模块的突变来讨论 ICE 封闭子类别的偏序集的 Hasse quiver。