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. Details. 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. Sketch of the proof. Let formula_1 be the category of left exact functors from the abelian category formula_2 to the category of abelian groups formula_3. First we construct a contravariant embedding formula_4 by formula_5 for all formula_6, where formula_7 is the covariant hom-functor, formula_8. The Yoneda Lemma states that formula_9 is fully faithful and we also get the left exactness of formula_9 very easily because formula_7 is already left exact. The proof of the right exactness of formula_9 is harder and can be read in Swan, "Lecture Notes in Mathematics 76 . After that we prove that formula_13 is an abelian category by using localization theory (also Swan). This is the hard part of the proof. It is easy to check that the abelian category formula_13 is an AB5 category with a generator formula_15. In other words it is a Grothendieck category and therefore has an injective cogenerator formula_16. The endomorphism ring formula_17 is the ring we need for the category of R"-modules. By formula_18 we get another contravariant, exact and fully faithful embedding formula_19 The composition formula_20 is the desired covariant exact and fully faithful embedding. Note that the proof of  the Gabriel–Quillen embedding theorem for exact categories is almost identical.