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- Mitchell's_embedding_theorem abstract "Mitchell's embedding theorem, also known as the Freyd–Mitchell 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 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.".
- Mitchell's_embedding_theorem wikiPageID "453755".
- Mitchell's_embedding_theorem wikiPageRevisionID "543753205".
- Mitchell's_embedding_theorem hasPhotoCollection Mitchell's_embedding_theorem.
- Mitchell's_embedding_theorem subject Category:Additive_categories.
- Mitchell's_embedding_theorem subject Category:Module_theory.
- Mitchell's_embedding_theorem subject Category:Theorems_in_algebra.
- Mitchell's_embedding_theorem type Abstraction100002137.
- Mitchell's_embedding_theorem type AdditiveCategories.
- Mitchell's_embedding_theorem type Class107997703.
- Mitchell's_embedding_theorem type Collection107951464.
- Mitchell's_embedding_theorem type Communication100033020.
- Mitchell's_embedding_theorem type Group100031264.
- Mitchell's_embedding_theorem type Message106598915.
- Mitchell's_embedding_theorem type Proposition106750804.
- Mitchell's_embedding_theorem type Statement106722453.
- Mitchell's_embedding_theorem type Theorem106752293.
- Mitchell's_embedding_theorem type TheoremsInAlgebra.
- Mitchell's_embedding_theorem comment "Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules.".
- Mitchell's_embedding_theorem label "Einbettungssatz von Mitchell".
- Mitchell's_embedding_theorem label "Mitchell's embedding theorem".
- Mitchell's_embedding_theorem sameAs Einbettungssatz_von_Mitchell.
- Mitchell's_embedding_theorem sameAs m.02bdr1.
- Mitchell's_embedding_theorem sameAs Q1148215.
- Mitchell's_embedding_theorem sameAs Q1148215.
- Mitchell's_embedding_theorem sameAs Mitchell's_embedding_theorem.
- Mitchell's_embedding_theorem wasDerivedFrom Mitchell's_embedding_theorem?oldid=543753205.
- Mitchell's_embedding_theorem isPrimaryTopicOf Mitchell's_embedding_theorem.