Category of modules

Template:Short description 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 $\mathbb{Z}$, 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.

Properties

The categories of left and right modules are abelian categories. These categories have enough projectives and enough injectives. 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.

Over a commutative ring, together with the tensor product of modules $\otimes$, 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 $R$.

A compact object in $R$-$𝐌𝐨𝐝$ is exactly a finitely presented module.

Category of vector spaces

The category $K\text{-}𝐕𝐞𝐜𝐭$ (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 $K$, $K\text{-}𝐕𝐞𝐜𝐭$ 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 $K\text{-}𝐕𝐞𝐜𝐭$. For example, the dimension theorem for vector spaces says that the isomorphism classes in $K\text{-}𝐕𝐞𝐜𝐭$ correspond exactly to the cardinal numbers, and that $K\text{-}𝐕𝐞𝐜𝐭$ is equivalent to the subcategory of $K\text{-}𝐕𝐞𝐜𝐭$ 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 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

External links

• Template:Ncatlab