Exactness of direct limits in the category of firm modules

Abstract

Let R be a nonunital ring. A left R-module M is said to be firm if R ⊗ R M → M given by r ⊗ m ↦ r m is an isomorphism. The category of firm left R-modules generalizes the usual category of unital modules for a unital ring but it is not abelian in general. In this paper we study the definition of exactness in the category of firm modules, we prove that direct limits are exact if the category is abelian and we also give an example of a ring in which direct limits are not exact in this category.