The category of firm modules need not be abelian

Let R be a nonunital ring. A left R-module M is said to be firm if R⊗RM→M given by r⊗m↦rm is an isomorphism. The category of firm left R-modules generalizes the usual category of unital modules for a unital ring and it has been used to study the Morita Theory for nonunital rings. It is an open problem if the category of firm modules is an abelian category. In this paper, we prove that, in general, this category is not abelian.