{{Short description|Category whose objects are R-modules and whose morphisms are module homomorphisms}} In [[Abstract algebra|algebra]], given a [[Ring (mathematics)|ring]] <math>R</math>, the '''category of left modules''' over <math>R</math> is the [[Category (mathematics)|category]] whose [[Object (category theory)|objects]] are all left [[Module (mathematics)|modules]] over <math>R</math> and whose [[morphism]]s are all [[module homomorphism]]s between left <math>R</math>-modules. For example, when <math>R</math> is the ring of [[integer]]s <math>\mathbb{Z}</math>, it is the same thing as the [[category of abelian groups]]. The '''category of right modules''' is defined in a similar way.
One can also define the category of [[bimodule]]s over a ring <math>R</math> but that category is equivalent to the category of left (or right) modules over the [[enveloping algebra of an associative algebra|enveloping algebra]] of <math>R</math> (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]].<ref>{{cite web|url=http://ncatlab.org/nlab/show/module+category|title=module category in nLab|work=ncatlab.org}}</ref>
== Properties == The categories of left and right modules are [[Abelian category|abelian categories]]. These categories have [[enough projectives]]<ref>trivially since any module is a quotient of a free module.</ref> and [[enough injectives]].<ref>{{harvnb|Dummit|Foote|loc=Ch. 10, Theorem 38.}}</ref> [[Mitchell's embedding theorem]] states every abelian category arises as a [[full subcategory]] of the category of modules over some ring.
[[Projective limit]]s and [[inductive limit]]s exist in the categories of left and right modules.<ref>{{harvnb|Bourbaki|loc=§ 6.}}</ref>
Over a [[commutative ring]], together with the [[tensor product of modules]] <math>\otimes</math>, the category of modules is a [[symmetric monoidal category]].
== Objects == {{expand section|date=March 2023}}
A [[monoid object]] of the category of modules over a commutative ring <math>R</math> is exactly an [[associative algebra]] over <math>R</math>.
A [[compact object (mathematics)|compact object]] in <math>R</math>-<math>\mathbf{Mod}</math> is exactly a finitely presented module.
== Category of vector spaces == {{see also|FinVect}} The [[category (mathematics)|category]] <math>K\text{-}\mathbf{Vect}</math> (some authors use <math>\mathbf{Vect}_K</math>) has all [[vector space]]s over a [[Field (mathematics)|field]] <math>K</math> as objects, and [[linear map|<math>K</math>-linear maps]] as morphisms. Since vector spaces over <math>K</math> (as a field) are the same thing as [[module (algebra)|module]]s over the [[ring (mathematics)|ring]] <math>K</math>, <math>K\text{-}\mathbf{Vect}</math> is a special case of <math>R</math>-<math>\mathbf{Mod}</math> (some authors use <math>\mathbf{Mod}_R</math>), the category of left <math>R</math>-modules.
Much of [[linear algebra]] concerns the description of <math>K\text{-}\mathbf{Vect}</math>. For example, the [[dimension theorem for vector spaces]] says that the [[isomorphism class]]es in <math>K\text{-}\mathbf{Vect}</math> correspond exactly to the [[cardinal number]]s, and that <math>K\text{-}\mathbf{Vect}</math> is [[equivalence of categories|equivalent]] to the [[subcategory]] of <math>K\text{-}\mathbf{Vect}</math> which has as its objects the vector spaces <math>K_n</math>, where <math>n</math> is any cardinal number.
== Generalizations == The category of [[sheaves of modules]] over a [[ringed space]] also has enough injectives (though not always enough projectives).
== See also == * [[Algebraic K-theory]] (the important invariant of the category of modules.) * [[Category of rings]] * [[Derived category]] * [[Module spectrum]] * [[Category of graded vector spaces]] * [[Category of representations]] * [[Change of rings]] * [[Morita equivalence]] * [[Stable module category]] * [[Eilenberg–Watts theorem]]
== References == {{reflist}}
===Bibliography=== *{{cite book |last=Bourbaki |author-link=Bourbaki group |title=Algèbre |chapter=Algèbre linéaire}} *{{cite book |last1=Dummit |first1=David |last2=Foote |first2=Richard |title=Abstract Algebra}} *{{cite book |first=Saunders |last=Mac Lane |authorlink=Saunders Mac Lane|title=[[Categories for the Working Mathematician]] | edition=second |date=September 1998 |publisher=Springer |isbn=0-387-98403-8 | zbl=0906.18001 | volume=5 | series=[[Graduate Texts in Mathematics]] }}
== External links == * {{ncatlab|id=Mod|title=Mod}}
[[Category:Categories in category theory|Vector spaces]] [[Category:Linear algebra]]