Inalgebra, given aring${\displaystyle R}$, thecategory of left modules over${\displaystyle R}$ is thecategory whoseobjects are all leftmodules over${\displaystyle R}$ and whosemorphisms are allmodule homomorphisms between left${\displaystyle R}$-modules. For example, when${\displaystyle R}$ is the ring ofintegers${\displaystyle \mathbb{Z}}$, it is the same thing as thecategory of abelian groups. Thecategory of right modules is defined in a similar way.

One can also define the category ofbimodules over a ring${\displaystyle R}$ but that category is equivalent to the category of left (or right) modules over theenveloping algebra of${\displaystyle R}$ (or over the opposite of that).

Note: Some authors use the termmodule category for the category of modules. This term can be ambiguous since it could also refer to a category with amonoidal-category action.^[1]

The categories of left and right modules areabelian categories. These categories haveenough projectives^[2] andenough injectives.^[3]Mitchell's embedding theorem states every abelian category arises as afull subcategory of the category of modules over some ring.

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

Over acommutative ring, together with thetensor product of modules${\displaystyle \otimes }$, the category of modules is asymmetric monoidal category.

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Amonoid object of the category of modules over a commutative ring${\displaystyle R}$ is exactly anassociative algebra over${\displaystyle R}$.

Acompact object in${\displaystyle R}$-${\displaystyle \mathbf{M}\mathbf{o}\mathbf{d}}$ is exactly a finitely presented module.

Thecategory${\displaystyle K\text{-}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}$ (some authors use${\displaystyle {\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_K}$) has allvector spaces over afield${\displaystyle K}$ as objects, and${\displaystyle K}$-linear maps as morphisms. Since vector spaces over${\displaystyle K}$ (as a field) are the same thing asmodules over thering${\displaystyle K}$,${\displaystyle K\text{-}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}$ is a special case of${\displaystyle R}$-${\displaystyle \mathbf{M}\mathbf{o}\mathbf{d}}$ (some authors use${\displaystyle {\mathbf{M}\mathbf{o}\mathbf{d}}_R}$), the category of left${\displaystyle R}$-modules.

Much oflinear algebra concerns the description of${\displaystyle K\text{-}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}$. For example, thedimension theorem for vector spaces says that theisomorphism classes in${\displaystyle K\text{-}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}$ correspond exactly to thecardinal numbers, and that${\displaystyle K\text{-}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}$ isequivalent to thesubcategory of${\displaystyle K\text{-}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}$ which has as its objects the vector spaces${\displaystyle K_n}$, where${\displaystyle n}$ is any cardinal number.

The category ofsheaves of modules over aringed space also has enough injectives (though not always enough projectives).

• 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

• 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.

• Mod at thenLab

• Categories in category theory
• Linear algebra

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