Modules With Descending Chain Condition

Eben Matlis
1960 Transactions of the American Mathematical Society  
Introduction. Throughout this discussion the ring £ will be a commutative, Noetherian ring with unit. The study of modules over such a ring has, heretofore, been largely confined to the study of finitely generated modules. The purpose of this paper is to introduce the study of modules with descending chain condition (D.C.C.), and their natural generalization-modules with maximal orders. Among the main tools in the study of these modules are the analysis of injective modules carried out in [4] ,
more » ... arried out in [4] , and the theory of duality for complete, local rings developed there. The results of the present paper guarantee the existence of a sufficient quantity of modules with D.C.C. and provide a basis for a link between the theory of such modules and the theory of finitely generated ones. The Koszul complex, with its dual nature, plays an important role in establishing this link. In §1 we introduce the functors X and XM-By considering these functors we are able to give characterizations of modules with maximal orders; and decompose them uniquely into direct sums, where each summand depends on only a single maximal ideal. We then prove a transition theorem which enables us to pass to rings of quotients and their completions. A key result of this section is the theorem that if A is an £-module with D.C.C, and if 2 is an ideal of £, then 74 =A if and only if there exists an element r£7 such that rA=A. This is the dual of a standard result for finitely generated mod-
doi:10.2307/1993385 fatcat:zdgejvllcnhzlgwl4ccftxdely