A representation theory for noetherian rings

Abstract

The representation theory of artinian rings has long been studied, and seldom more intensely than in the past few years. However, with the exception of integral representations of finite groups, there is no representation theory for noetherian rings even when they are commutative. Our aim in this paper is to develop such a theory. We attempt to pattern it after the known representation theory of artinian rings. For this, we require a notion of indecomposability different from the usual one; and we say that a module is strongly indecomposable if it is noetherian, and no factor by a nontrivial direct sum of submodules has lower Krull dimension than the module. Thus strongly indecomposable modules are indecomposable and, by way of example, any uniform noetherian module is strongly indecomposable. Our first result states that some factor of a noetherian module by a direct sum of strongly indecomposable submodules has dimension less than the dimension of the module. This decomposition refines indecomposable noetherian modules, but we can say little about it without restricting the underlying ring. Thus we study, in Section 1, or-indecomposable that is, strongly indecomposable a-dimensional modules that have no nonzero submodule of dimension < cz. The resemblance of these to indecomposable modules of finite length is analogous to that of a-critical modules to simple ones (see [8]). Of course a-indecomposables have the disadvantage that, in general, not all of the strongly indecomposable submodules in the decomposition just described can be chosen to be ol-indecomposable for some 01. But each factor in the submodule sequence of a noetherian module (see Section 2) can be decomposed in terms of them; and the or-indecomposables appearing in such a decomposition have certain envelopes, typically proper submodules