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.

**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] }

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.

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.

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**, 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 *K*^{n}, where *n* is any cardinal number.

The category of sheaves of modules over a ringed 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 abelian groups
- Category of representations

In mathematics, an **abelian category** is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, **Ab**. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very *stable* categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.

In mathematics, specifically in category theory, a **preadditive category** is another name for an **Ab-category**, i.e., a category that is enriched over the category of abelian groups, **Ab**. That is, an **Ab-category****C** is a category such that every hom-set Hom(*A*,*B*) in **C** has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas:

In abstract algebra, the endomorphisms of an abelian group *X* form a ring. This ring is called the **endomorphism ring***X*, denoted by End(*X*); the set of all homomorphisms of *X* into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity.

In mathematics, specifically in category theory, an **additive category** is a preadditive category **C** admitting all finitary biproducts.

In category theory, a branch of mathematics, the concept of an **injective cogenerator** is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation, and (dually) **cogenerators** are objects which envelope other objects as an approximation.

In mathematics, a **module** is one of the fundamental algebraic structures used in abstract algebra. A **module over a ring** is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring and a multiplication is defined between elements of the ring and elements of the module. A module taking its scalars from a ring *R* is called an *R*-module.

In mathematics, a **monoidal category** is a category **C** equipped with a bifunctor

The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of *objects* and *arrows*, where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

**Mitchell's embedding theorem**, also known as the **Freyd–Mitchell theorem** or the **full embedding theorem**, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.

In mathematics, especially in the field of category theory, the concept of **injective object** is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.

In abstract algebra, a **bimodule** is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules.

In mathematics, especially in category theory, a **closed monoidal category** is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.

This is a glossary of properties and concepts in category theory in mathematics.

In category theory, the notion of a **projective object** generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object.

In category theory, a branch of mathematics, a **monoid** in a monoidal category is an object *M* together with two morphisms

In algebra, the **zero object** of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as {0}. One often refers to *the* trivial object since every trivial object is isomorphic to any other.

In mathematics, the **category of rings**, denoted by **Ring**, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

In mathematics, a **Grothendieck category** is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.

In mathematics, the **quotient** of an abelian category by a Serre subcategory * is the abelian category ** which, intuitively, is obtained from ** by ignoring all objects from **.* There is a canonical exact functor whose kernel is *.*

In representation theory, the **category of representations** of some algebraic structure `A` has the representations of `A` as objects and equivariant maps as morphisms between them. One of the basic thrusts of representation theory is to understand the conditions under which this category is semisimple; i.e., whether an object decomposes into simple objects.

- ↑ "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.

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

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

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.