An Axiomatic Description of a Duality for Modules

Abstract

Take for instance a k-algebra 4 over a field k. Then the usual k-dual DM=Homk(M, k) of a finite dimensional 4-module M satisfies all three properties. However for infinite dimensional 4-modules the condition (D1) does not hold, and for most rings there is even on the level of the finite length modules no duality functor between right and left modules available. The aim of this note is to describe without any restriction on the ring 4 a bijection between certain classes of right and left 4-modules which satisfies (D1) (D3) and which is uniquely determined by these properties. This bijection covers the k-duality mentioned above but usually there will be non-finitely generated modules on which D is defined. Our approach gives a new interpretation of a construction which is known as elementary duality amongst model theorists, and which is due to Herzog [6], see also [9]. We formulate now the main result. To this end denote by Mod(4) the category of (right) 4-modules and by mod(4) the full subcategory of all finitely presented 4-modules. We shall identify the category of left 4-modules with Mod(4). Recall that a 4-module M is pure-injective if every pure monomorphism starting in M splits. A map .: M N is a pure monomorphism if . 4 X : M 4 X N 4 X is a monomorphism for every X in mod(4). We call an indecomposable pure-injective 4-module article no. AI971660