  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, that ICE-closed
subcategories are precisely torsion classes in some wide subcategories, 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 Schr\"oder number for type A case.
