# Category of modules > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Category_of_modules > Markdown URL: https://mediated.wiki/source/Category_of_modules.md > Source: https://en.wikipedia.org/wiki/Category_of_modules > Source revision: 1347375233 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Category whose objects are R-modules and whose morphisms are module homomorphisms In [algebra](/source/Abstract_algebra), given a [ring](/source/Ring_(mathematics)) R {\displaystyle R} , the **category of left modules** over R {\displaystyle R} is the [category](/source/Category_(mathematics)) whose [objects](/source/Object_(category_theory)) are all left [modules](/source/Module_(mathematics)) over R {\displaystyle R} and whose [morphisms](/source/Morphism) are all [module homomorphisms](/source/Module_homomorphism) between left R {\displaystyle R} -modules. For example, when R {\displaystyle R} is the ring of [integers](/source/Integer) Z {\displaystyle \mathbb {Z} } , it is the same thing as the [category of abelian groups](/source/Category_of_abelian_groups). The **category of right modules** is defined in a similar way. One can also define the category of [bimodules](/source/Bimodule) over a ring R {\displaystyle R} but that category is equivalent to the category of left (or right) modules over the [enveloping algebra](/source/Enveloping_algebra_of_an_associative_algebra) of R {\displaystyle R} (or over the opposite of that). **Note:** Some authors use the term **module category** for the category of modules. This term can be ambiguous since it could also refer to a category with a [monoidal-category action](/source/Monoidal-category_action).[1] ## Properties The categories of left and right modules are [abelian categories](/source/Abelian_category). These categories have [enough projectives](/source/Enough_projectives)[2] and [enough injectives](/source/Enough_injectives).[3] [Mitchell's embedding theorem](/source/Mitchell's_embedding_theorem) states every abelian category arises as a [full subcategory](/source/Full_subcategory) of the category of modules over some ring. [Projective limits](/source/Projective_limit) and [inductive limits](/source/Inductive_limit) exist in the categories of left and right modules.[4] Over a [commutative ring](/source/Commutative_ring), together with the [tensor product of modules](/source/Tensor_product_of_modules) ⊗ {\displaystyle \otimes } , the category of modules is a [symmetric monoidal category](/source/Symmetric_monoidal_category). ## Objects This section needs expansion. You can help by adding missing information. (March 2023) A [monoid object](/source/Monoid_object) of the category of modules over a commutative ring R {\displaystyle R} is exactly an [associative algebra](/source/Associative_algebra) over R {\displaystyle R} . A [compact object](/source/Compact_object_(mathematics)) in R {\displaystyle R} - M o d {\displaystyle \mathbf {Mod} } is exactly a finitely presented module. ## Category of vector spaces See also: [FinVect](/source/FinVect) The [category](/source/Category_(mathematics)) K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } (some authors use V e c t K {\displaystyle \mathbf {Vect} _{K}} ) has all [vector spaces](/source/Vector_space) over a [field](/source/Field_(mathematics)) K {\displaystyle K} as objects, and [K {\displaystyle K} -linear maps](/source/Linear_map) as morphisms. Since vector spaces over K {\displaystyle K} (as a field) are the same thing as [modules](/source/Module_(algebra)) over the [ring](/source/Ring_(mathematics)) K {\displaystyle K} , K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } is a special case of R {\displaystyle R} - M o d {\displaystyle \mathbf {Mod} } (some authors use M o d R {\displaystyle \mathbf {Mod} _{R}} ), the category of left R {\displaystyle R} -modules. Much of [linear algebra](/source/Linear_algebra) concerns the description of K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } . For example, the [dimension theorem for vector spaces](/source/Dimension_theorem_for_vector_spaces) says that the [isomorphism classes](/source/Isomorphism_class) in K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } correspond exactly to the [cardinal numbers](/source/Cardinal_number), and that K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } is [equivalent](/source/Equivalence_of_categories) to the [subcategory](/source/Subcategory) of K - V e c t {\displaystyle K{\text{-}}\mathbf {Vect} } which has as its objects the vector spaces K n {\displaystyle K_{n}} , where n {\displaystyle n} is any cardinal number. ## Generalizations The category of [sheaves of modules](/source/Sheaves_of_modules) over a [ringed space](/source/Ringed_space) also has enough injectives (though not always enough projectives). ## See also - [Algebraic K-theory](/source/Algebraic_K-theory) (the important invariant of the category of modules.) - [Category of rings](/source/Category_of_rings) - [Derived category](/source/Derived_category) - [Module spectrum](/source/Module_spectrum) - [Category of graded vector spaces](/source/Category_of_graded_vector_spaces) - [Category of representations](/source/Category_of_representations) - [Change of rings](/source/Change_of_rings) - [Morita equivalence](/source/Morita_equivalence) - [Stable module category](/source/Stable_module_category) - [Eilenberg–Watts theorem](/source/Eilenberg%E2%80%93Watts_theorem) ## References 1. **[^](#cite_ref-1)** ["module category in nLab"](http://ncatlab.org/nlab/show/module+category). *ncatlab.org*. 1. **[^](#cite_ref-2)** trivially since any module is a quotient of a free module. 1. **[^](#cite_ref-3)** [Dummit & Foote](#CITEREFDummitFoote), Ch. 10, Theorem 38. 1. **[^](#cite_ref-4)** [Bourbaki](#CITEREFBourbaki), § 6. ### Bibliography - [Bourbaki](/source/Bourbaki_group). "Algèbre linéaire". *Algèbre*. - Dummit, David; Foote, Richard. *Abstract Algebra*. - [Mac Lane, Saunders](/source/Saunders_Mac_Lane) (September 1998). *[Categories for the Working Mathematician](/source/Categories_for_the_Working_Mathematician)*. [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics). Vol. 5 (second ed.). Springer. [ISBN](/source/ISBN_(identifier)) [0-387-98403-8](https://en.wikipedia.org/wiki/Special:BookSources/0-387-98403-8). [Zbl](/source/Zbl_(identifier)) [0906.18001](https://zbmath.org/?format=complete&q=an:0906.18001). ## External links - [Mod](https://ncatlab.org/nlab/show/Mod) at the [*n*Lab](/source/NLab) --- Adapted from the Wikipedia article [Category of modules](https://en.wikipedia.org/wiki/Category_of_modules) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Category_of_modules?action=history)). 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