Classifying subcategories of modules over a Noetherian ring

Let R be a commutative ring with unit. The main topic in this paper is the classification of certain subcategories of R-modules and subcategories of the derived category of R -modules through order preserving (reversing) bijections between the set. of such subcategories and the set of certain subsets of the prime spectrum and Ziegler spectrum of R. We consider Serre subcategories and wide subcategories of R-modules or finitely generated R-modules as well as thick subcategories and localizing subcategories of the derived subcategory D(R). We give a brief proof of the correspondence between subsets of Spec(R) which can he written as a union of closed subsets and Serre subcategories of finitely generated R-modules under the assumption that R is Noetherian. If we define a narrow subcategory of an abelian category as a full subcategory closed under extensions and cokernels, we can generalize this result to a bijection between such subsets of Spec(R) and the set of narrow subcategories of finitely generated R-modules with a further condition that R has finite Krull dimension.