The Freyd-Mitchell embedding theorem states the following:

Let $\mathcal{A}$ be a small abelian category. There exists a unital ring $R$ and a full, faithful and exact functor $F\colon\mathcal{A}\rightarrow R\mathcal{Mod}$, where $R\mathcal{Mod}$ is the (abelian) category of (left) $R$-modules.

**I am searching for (elementary and non-elementary, useful and non-useful) applications of this theorem. I am also interested in applications, for which one does not necessarly need the embedding theorem.**

The only "real" application I know at the moment is all about diagram chasing:

Checking exactness of diagrams in arbitrary abelian categories can be done by checking those in the categories of $R$-modules, where one can apply element-wise diagram chasing.

(This can be made precise, what is e.g. done in Freyd's "Abelian Categories - An Introduction to the Theory of Functors".)

a prioriexplanation of why certain contructions that are obvious and intuitive in $R$-Mod can be performed in any abelian category. Since I for one find the basic formalism of abelian categories somewhat tedious, I do appreciate the result for this reason. – Pete L. Clark Apr 21 '13 at 1:00