Remarks on subcategories of Artinian modules

Abstract

We study two subcategories of the category of Artinian modules, a wide subcategory and a Serre subcategory. We prove that all wide subcategories of Artinian modules are Serre subcategories. We also provide the bijection between the set of Serre subcategories and the set of specialization closed subsets of the set of closed prime ideals of some completed ring. These results are Artinian analogues of the theorems proved by Takahashi.