A useful method in the study of an Abelian group is consideration of its completions with respect to various popular topologies, e.g., p-adic topologies. A flaw in this method is that it is not always applicable; a group may fail to be Hausdorff and thus have no completion. In this paper we define the “completion” of a group G (more generally, of a module over a Dedekind ring R) as Ext(Q/R, G), and we observe that this construction has properties reminiscent of metric completions. These results are really folklore, based on results of Harrison [j’] and Nunke [9] (see also Matlis [S]), and are compiled in Section 2. In the next section, axioms are given for the module Ext(Q/R, G) and for the functor Ext(Q/R, ). Section 4 makes the topological analogy precise by furnishing modules with a topology (in which, unfortunately, addition need not be jointly continuous, but which is TX if and only if the module is reduced). The “completions” defined algebraically as Ext’s are exactly those modules which are complete in the sense that they are closed whenever they are imbedded as submodules. The next section gives some applications to groups. The final section shows that if one completes the underlying R-module of an R-algebra (where R is Dedekind), the resulting completion is an R-algebra. For a large class of algebras, the completion is faithfully flat.