# Mitchell's embedding theorem

**Mitchell's embedding theorem**, also known as the **Freyd–Mitchell theorem** or the **full embedding theorem**, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.

The precise statement is as follows: if **A** is a small abelian category, then there exists a ring *R* (with 1, not necessarily commutative) and a full, faithful and exact functor *F*: **A** → *R*-Mod (where the latter denotes the category of all left *R*-modules).

The functor *F* yields an equivalence between **A** and a full subcategory of *R*-Mod in such a way that kernels and cokernels computed in **A** correspond to the ordinary kernels and cokernels computed in *R*-Mod. Such an equivalence is necessarily additive.
The theorem thus essentially says that the objects of **A** can be thought of as *R*-modules, and the morphisms as *R*-linear maps, with kernels, cokernels, exact sequences and sums of morphisms being determined as in the case of modules. However, projective and injective objects in **A** do not necessarily correspond to projective and injective *R*-modules.

Note that the proof of the Gabriel–Quillen embedding theorem for exact categories is almost identical.