Category of modules
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
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 over 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
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A monoid object of the category of modules over a commutative ring R is exactly an associative algebra over R.
A compact object in R-Mod is exactly a finitely presented module.
Category of vector spaces
The category K-Vect (some authors use VectK) 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 ModR), 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
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
- ^ "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
- 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.