Lifting idempotents also has major consequences for the category of R modules.

This same construction may be carried out if the A < sub > i </ sub >' s are sets, rings, modules ( over a fixed ring ), algebras ( over a fixed field ), etc., and the homomorphisms are homomorphisms in the corresponding category.

In the language of category theory it is a morphism in the category of modules over a given ring.

Since plug-ins and extensions both increase the utility of the original application, Mozilla uses the term " add-on " as an inclusive category of augmentation modules that consists of plug-ins, themes, and search engines.

For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.

This is formalized by studying the Ext functor and describing the module category in various ways including quivers ( whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2 ) and Auslander – Reiten theory where the associated graph has a vertex for every indecomposable module.

The ring takes the place of the category C, and the category of modules over the ring is a category of functors defined on C.

* If R is a ring, then the category of all left ( or right ) modules over R is an abelian category.

In fact, it can be shown that any abelian category is equivalent to a full subcategory of such a category of modules ( Mitchell's embedding theorem ).

* If R is a left-noetherian ring, then the category of finitely generated left modules over R is abelian.

In particular, the category of finitely generated modules over a noetherian commutative ring is abelian ; in this way, abelian categories show up in commutative algebra.

There are zero objects also for the category of abelian groups, category of pseudo-rings Rng ( trivial ring ), category of modules over a ring, and category of vector spaces over a field ; see zero object ( algebra ) for details.

* The category of ( left ) modules over a ring R, in particular:

The latter example leads to a generalization of modules over rings: If C is a preadditive category, then Mod ( C ) := Add ( C, Ab ) is called the module category over C. When C is the one-object preadditive category corresponding to the ring R, this reduces to the ordinary category of ( left ) R-modules.

In a category with exponentials, using the isomorphism ( in computer science, this is called currying ), the Ackermann function may be defined via primitive recursion over higher-order functionals as follows:

Dual vector space: The map which assigns to every vector space its dual space and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

The inflectional categories of the Greek verb have likewise remained largely the same over the course of the language's history, though with significant changes in the number of distinctions within each category and their morphological expression.

As with category C, every Austrian citizen at the age of 18 or over can buy and own this type of weapon without further registration or permission, since no background checks are performed also EU / EEA citizens can buy them without problems.

In a study of individuals diagnosed with " severe mental illness " living in a US inner-city area, a quarter were found to have been victims of at least one violent crime over the course of a year, a proportion eleven times higher than the inner-city average, and higher in every category of crime including violent assaults and theft.

The list of ' modern ' world champions extended into the 1980s, when the sport entered the catamaran, and then the ' superboat ' era-the 1000 cubic inch total engine displacement restrictions were lifted for boats over in length, and soon three-and four-engine boats sporting F16 fighter canopies replaced the venerable vee hulls that had been the sport's top category for twenty years.

Alternately, a preordered set can be understood as an enriched category, enriched over the category 2

Grand theft generally consists of the theft of something of value over $ 1000 ( it can be money, labor or property but is lower with respect to various specified property ), while petty theft is the default category for all other thefts.

Let C be the category of vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K ( assumed to be unital and associative ).

100BaseVG is a 100 Mbit / s Ethernet standard specified to run over four pairs of category 3 UTP wires ( known as voice grade, hence the " VG ").

The notion of a natural transformation is categorical, and states ( informally ) that a particular map between functors can be done consistently over an entire category.

* A category is preadditive if it is enriched over the monoidal category Ab of abelian groups.

IFVs are usually tracked, but some wheeled vehicles fall into this category, too.

" Some of the dishes included in the category of minutas are milanesas, churrascos, bifes, escalopes, tallarines, ravioles ( ravioli ), ñoquis ( gnocchi, although some are very typical of locations that sell food: " bifes a caballo " ( beef steak with two fried eggs ), " milanesa a caballo ", " milanesa completa " ( a milanesa with two fried eggs and a garnish of fries ), " revuelto Gramajo ", " colchón de arvejas ", " suprema de pollo " ( a kind of chicken milanesa ), matambres, " lengua a la vinagreta " and " sandwiches " ( sandwiches de miga ).

The interpretation of an n-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original n-ary morphism will work in a monoidal category.

For some schools of Hinduism and Buddhism the received textual tradition is an epistemological category equal to perception and inference ( although this is not necessarily true for some other schools ).

Some of these sites include fashion jewelry as a separate category, while some use this term in favor of costume jewelry.

Some strips which are still in affiliation with the original creator are produced by small teams or entire companies, such as Jim Davis ' Garfield, however there is some debate if these strips fall in this category.

In applying the elemental theory to beings that function on a cosmic scale ( e. g. Yog-Sothoth ) some authors created a separate category termed aethyr.

Abstracting again, a category is itself a type of mathematical structure, so we can look for " processes " which preserve this structure in some sense ; such a process is called a functor.

At the very least, category theoretic language clarifies what exactly these related areas have in common ( in some abstract sense ).

These categories surely have some objects that are " special " in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i. e., we do not know whether an object A is a set, a topology, or any other abstract concept – hence, the challenge is to define special objects without referring to the internal structure of those objects.

Among the conditions purportedly treated were arthritis, allergies, asthma, some coronary difficulties, eye trouble, ulcers, migraine headaches, ' sexual deviation ' ( a category which for Hubbard included homosexuality ) and even death.

Documentary films constitute a broad category of nonfictional motion pictures intended to document some aspect of reality, primarily for the purposes of instruction or maintaining a historical record.

* Dual ( grammatical number ), a grammatical category used in some languages

He suggests that one element of this expansion in use " involves the application of the term diaspora to an ever-broadening set of cases: essentially to any and every nameable population category that is to some extent dispersed in space ".

This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below.

Most small arms and some pyrotechnic items fall into this category.

Finally, some theorists place emotions within a more general category of " affective states " where affective states can also include emotion-related phenomena such as pleasure and pain, motivational states ( for example, hunger or curiosity ), moods, dispositions and traits.

According to the WHO, some practices regarded as legal in countries that have outlawed FGM do fall under the category of Type IV ( see below ), but the organization decided to maintain a broad definition to avoid loopholes that could allow FGM to continue.

There is some overlap between the terms, particularly in that some moderate-sized vegetable growing concerns, often called market gardening, can fit in either category.

This origin gave rise to a wide variety of dishes called gazpacho and to some others that don't have this name but belong to the same category.

While clear categories do exist, it is sometimes difficult to choose which category some individual heads of state belong to.

Often depending on which constitutional category ( above ) a head of state belongs to, they may have some or all of the roles listed below, and various other ones.

The much more numerous category of vici, " small towns " grew on informal plans, often round a camp or at a ford or crossroads ; some were not small, others were scarcely urban, some not even defended by a wall, the characteristic feature of a place of any importance.