Rigid modules and ICE-closed subcategories over path algebras

We introduce image-cokernel-extension-closed (ICE-closed) subcategories of module categories. This class unifies both torsion classes and wide subcategories. We show that ICE-closed subcategories over the path algebra of Dynkin type are in bijection with basic rigid modules, and that the number does not depend on the orientation of the quiver. We give an explicit formula of this number for each Dynkin type, and in particular, it is equal to the large Schroder number for type A case.