Rigid modules and ICE-closed subcategories in quiver representations.

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 ICE-closed subcategories are precisely torsion classes in some wide subcategories. We also study natural maps from ICE-closed subcategories to torsion classes and wide subcategories in terms of rigid modules. Finally, we prove that the number of ICE-closed subcategories does not depend on the orientation of the quiver, and give an explicit formula for each Dynkin type, which is equal to the large Schröder number for type A. [ABSTRACT FROM AUTHOR]