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 Rmodules. 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 monoidalcategory action.^{[1]}
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Working with modules

8 Proj, Inj and Flat Modules

Lecture 1.2: Dmodules and quasicoherent sheaves on prestacks (N. Rozenblyum)
Transcription
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
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 Rmod is exactly a finitely presented module).
Category of vector spaces
The category KVect (some authors use Vect_{K}) has all vector spaces over a field K as objects, and Klinear maps as morphisms. Since vector spaces over K (as a field) are the same thing as modules over the ring K, KVect is a special case of RMod (some authors use Mod_{R}), the category of left Rmodules.
Much of linear algebra concerns the description of KVect. For example, the dimension theorem for vector spaces says that the isomorphism classes in KVect correspond exactly to the cardinal numbers, and that KVect is equivalent to the subcategory of KVect which has as its objects the vector spaces K^{n}, 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 Ktheory (the important invariant of the category of modules.)
 Category of rings
 Derived category
 Module spectrum
 Category of graded vector spaces
 Category of abelian groups
 Category of representations
 Change of rings
 Morita equivalence
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 0387984038. Zbl 0906.18001.
External links