On the exactness of the completion functor

Abstract

Suppose R is a commutative Noeheian ring with maximal ideal P and E is the injective hull of R/P. The hereditary torsion theory associated with P has E has as an injective cogenerator and the associated localization is exact. Matlis [3] has shown that E Artinian. In this case it can be shown that the functor associated wity localization and then completions is exact on finitely generated modules. In this paper we examine completions of modules over noncommutative rings. We show that if an injective congenerator for a torsion theory is Artinian and the associated localization is exact then the functor given by localization and then completion is also exact. Throughout this paper we will assume that all rings are associative with identity. If R is a ring we denote RM as the category of all unital left R-modules. We will adopt the convention of writing all module homomorphisms opposite their scalars. We refer the reader to [5] and [1] for the basic results about torsion theories and kernel functors.