On the Noetherian property in endomorphism rings of certain comodules

Abstract

In the above (and throughout this paper) Noetherian means right and left Noetherian and finite dimensional means of finite k dimension. By an R comodule we understand a left R comodule, that is, a pair (V, r) where V is a k-space and r: V-+R Ok V is a k map satisfying (1 @ 7)7= Q @ 1)7, (E @ I)7 = 1, the identity map. The terms injective and essential extension have the appropriate meanings in the category of left comodules. We refer the reader to [7] for the basic representation theory of comodules. All rings in this paper have an identity and we insist that the identity of a subring coincides with that of the overring. This work arose from question about the representation theory, in characteristic zero, of polycyclic-by-finite groups: the reader who feels that practical examples are lacking in the present work is advised to consult the companion paper [5]. Our results also apply to representations of finite dimensional Lie algebras and rational representations of algebraic groups in characteristic zero. At the heart of our proof of Theorem A lie certain results on completion in noncommutative Noetherian rings. These are dealt with in the first three 394 002 l-8693/8 l/060394-26%02.00/O