Fbn To Ring Module Categories Subcategory Classification

In this article,we generalize the notions such as support of modules and coherent subsets of Spec(R) to noncommutative noetherian rings.And by investigating the structure of indecomposable injective modules,we gain some maps between subcategories of Mod-R and subsets of Spec(R) by taking associated prime ideas and support of modules over FBN rings.For the former,we characterize the ring when there is a bijection between the set of full subcategories of Mod-R,which are closed under taking submodules, extensions,and direct unions,and the set of subsets of Spec(R).While for the latter,we give a sufficient condition to insure there is a bijection between the set of full subcategories of Mod-R,which are thick and closed under taking direct sums,and the set of coherent subsets of Spec(R).These conclusions can be seen as generalizations of main results of[5].