Category of modules: Difference between revisions

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In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, 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 R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of 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.^[1]

Properties[edit]

The categories of left and right modules are abelian categories. These categories have enough projectives^[2] and enough injectives.^[3] Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules of some ring.

Projective limits and inductive limits exist in the categories of left and right modules.^[4]

Over a commutative ring, together with the tensor product of modules ⊗, the category of modules is a symmetric monoidal category.

Objects[edit]

A monoid object of the category of modules over a commutative ring R is exactly an associative algebra over R.

See also: compact object (a compact object in the R-mod is exactly a finitely presented module).

Category of vector spaces[edit]

The category K-Vect (some authors use Vect_K) has all vector spaces over a field K as objects, and K-linear maps as morphisms. Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod (some authors use Mod_R), the category of left R-modules.

Much of linear algebra concerns the description of K-Vect. For example, the dimension theorem for vector spaces says that the isomorphism classes in K-Vect correspond exactly to the cardinal numbers, and that K-Vect is equivalent to the subcategory of K-Vect which has as its objects the vector spaces Kⁿ, where n is any cardinal number.

Generalizations[edit]

The category of sheaves of modules over a ringed space also has enough injectives (though not always enough projectives).

References[edit]

^ "module category in nLab". ncatlab.org.
^ trivially since any module is a quotient of a free module.
^ Dummit & Foote, Ch. 10, Theorem 38.
^ Bourbaki, § 6.

Bibliography[edit]

Bourbaki. "Algèbre linéaire". Algèbre.
Dummit, David; Foote, Richard. Abstract Algebra.
Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (second ed.). Springer. ISBN 0-387-98403-8. Zbl 0906.18001.

External links[edit]

http://ncatlab.org/nlab/show/Mod

This linear algebra-related article is a stub. You can help Wikipedia by expanding it.

[1] "module category in nLab". ncatlab.org.

[2] trivially since any module is a quotient of a free module.

[3] Dummit & Foote, Ch. 10, Theorem 38.

[4] Bourbaki, § 6.

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
-{{Histmerge|Draft:Category of modules}}
+{{Short description|Category in mathematics}}
-In [[Abstract algebra|algebra]], given a [[ring (mathematics)|ring]] ''R'', the '''category of left modules''' over ''R'' is the [[Category (mathematics)|category]] whose objects are all left [[module (mathematics)|modules]] over ''R'' and whose morphisms are all [[module homomorphism]]s between left ''R''-modules. For example, when ''R'' is the ring of integers '''Z''', it is the same thing as the [[category of abelian groups]]. The '''category of right modules''' is defined in a similar way.
+In [[Abstract algebra|algebra]], given a [[Ring (mathematics)|ring]] ''R'', the '''category of left modules''' over ''R'' is the [[Category (mathematics)|category]] whose [[Object (category theory)|objects]] are all left [[Module (mathematics)|modules]] over ''R'' and whose [[morphism]]s are all [[module homomorphism]]s between left ''R''-modules. For example, when ''R'' is the ring of [[integer]]s '''Z''', 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 ''R'' but that category is equivalent to the category of left (or right) modules over the [[enveloping algebra of an associative algebra|enveloping algebra]] of ''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]].<ref>{{cite web|url=http://ncatlab.org/nlab/show/module+category|title=module category in nLab|work=ncatlab.org}}</ref>
 == Properties ==
-The category of left modules (or that of right modules) is an [[abelian category]]. The category has [[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.
+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 of some ring.
-[[Projective limit]]s and [[inductive limit]]s exist in the category of (say left) modules.<ref>{{harvnb|Bourbaki|loc=§ 6.}}</ref>
+[[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]] ⊗, the category of modules is a [[symmetric monoidal category]].
+Over a [[commutative ring]], together with the [[tensor product of modules]] ⊗, the category of modules is a [[symmetric monoidal category]].
+== Objects ==
-== Example: the category of vector spaces ==
+{{expand section|date=March 2023}}
-The [[category (mathematics)|category]] '''K-Vect''' (some authors use '''Vect'''<sub>''K''</sub>) has all [[vector space]]s over a fixed [[Field (mathematics)|field]] ''K'' as [[object (category theory)|objects]] and [[linear transformation|''K''-linear transformations]] as [[morphism]]s. Since vector spaces over ''K'' (as a field) are the same thing as [[module (algebra)|module]]s over the [[ring (mathematics)|ring]] ''K'', '''K-Vect''' is a special case of '''R-Mod''', the category of left ''R''-modules.
+A [[monoid object]] of the category of modules over a commutative ring ''R'' is exactly an [[associative algebra]] over ''R''.
-Much of [[linear algebra]] concerns the description of '''K-Vect'''. For example, the [[dimension theorem for vector spaces]] says that the [[isomorphism class]]es in '''K-Vect''' correspond exactly to the [[cardinal number]]s, and that '''K-Vect''' is [[equivalence of categories|equivalent]] to the subcategory of '''K-Vect''' which has as its objects the free vector spaces ''K''<sup>''n''</sup>, where ''n'' is any cardinal number.
+See also: [[compact object (mathematics)|compact object]] (a compact object in the ''R''-mod is exactly a finitely presented module).
+== Category of vector spaces ==
+{{see also|FinVect}}
+The [[category (mathematics)|category]] ''K''-'''Vect''' (some authors use '''Vect'''<sub>''K''</sub>) has all [[vector space]]s over a [[Field (mathematics)|field]] ''K'' as objects, and [[linear map|''K''-linear maps]] as morphisms. Since vector spaces over ''K'' (as a field) are the same thing as [[module (algebra)|module]]s over the [[ring (mathematics)|ring]] ''K'', ''K''-'''Vect''' is a special case of ''R''-'''Mod''' (some authors use '''Mod'''<sub>''R''</sub>), the category of left ''R''-modules.
+Much of [[linear algebra]] concerns the description of ''K''-'''Vect'''. For example, the [[dimension theorem for vector spaces]] says that the [[isomorphism class]]es in ''K''-'''Vect''' correspond exactly to the [[cardinal number]]s, and that ''K''-'''Vect''' is [[equivalence of categories|equivalent]] to the [[subcategory]] of ''K''-'''Vect''' which has as its objects the vector spaces ''K''<sup>''n''</sup>, where ''n'' is any cardinal number.
 == Generalizations ==
@@ Line 20: / Line 30: @@
 == See also ==
-*[[Algebraic K-theory]] (the important invariant of the category of modules.)
+* [[Algebraic K-theory]] (the important invariant of the category of modules.)
-*[[Category of rings]]
+* [[Category of rings]]
-*[[Derived category]]
+* [[Derived category]]
-*[[Module spectrum]]
+* [[Module spectrum]]
 * [[Category of graded vector spaces]]
 * [[Category of abelian groups]]
 * [[Category of representations]]
+* [[Change of rings]]
+* [[Morita equivalence]]
 == References ==
 {{reflist}}
-*Bourbaki, ''Algèbre''; "Algèbre linéaire."
+===Bibliography===
-*Dummit, David; Foote, Richard. ''Abstract Algebra''.
+*{{cite book |last=Bourbaki |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]] }}
@@ Line 41: / Line 55: @@
-{{algebra-stub}}
+{{linear-algebra-stub}}