{{Short description|Category whose objects are R-modules and whose morphisms are module homomorphisms}} In algebra, given a ring <math>R</math>, the '''category of left modules''' over <math>R</math> is the category whose objects are all left modules over <math>R</math> and whose morphisms are all module homomorphisms between left <math>R</math>-modules. For example, when <math>R</math> is the ring of integers <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 bimodules over a ring <math>R</math> but that category is equivalent to the category of left (or right) modules over the 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 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 limits and inductive limits 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 in <math>R</math>-<math>\mathbf{Mod}</math> is exactly a finitely presented module.
== Category of vector spaces == {{see also|FinVect}} The category <math>K\text{-}\mathbf{Vect}</math> (some authors use <math>\mathbf{Vect}_K</math>) has all vector spaces over a field <math>K</math> as objects, and <math>K</math>-linear maps as morphisms. Since vector spaces over <math>K</math> (as a field) are the same thing as modules over the 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 classes in <math>K\text{-}\mathbf{Vect}</math> correspond exactly to the cardinal numbers, and that <math>K\text{-}\mathbf{Vect}</math> is 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}}
Vector spaces Category:Linear algebra