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#REDIRECT [[Subobject#quotient object]] {{R to anchor}} [[Category:Quotient objects| ]] ...

#REDIRECT [[Equivalence class#quotient set]] [[Category:Quotient objects|Set]] ...

{{for|the quotient of an [[abelian category]] by a [[Serre subcategory]]|Quotient of an abelian category}} ...gories]], analogous to a [[quotient group]] or [[Quotient space (topology)|quotient space]], but in the categorical setting. ...

...(ring theory)|ideal]], not a [[subring]], and a [[quotient group]] is the quotient of a group by a [[normal subgroup]], not by a general [[subgroup]]). ...and a submodule {{mvar|B}} of {{mvar|A}}, the [[Quotient space (topology)|quotient space]] {{math|''A''/''B''}} is defined by the [[equivalence relation]] ...

...(topology)|subspaces]] from [[topology]]. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a [ ...]]s, [[quotient group]]s, [[quotient space (topology)|quotient spaces]], [[quotient graph]]s, etc. ...

...ort description|Generalization of a quotient by an equivalence relation to objects in an arbitrary category}} ...alization of a [[quotient set|quotient]] by an [[equivalence relation]] to objects in an arbitrary [[category (mathematics)|category]]. It is the categorical ...

*Objects named for Austrian mathematician [[Emil Artin]] (1898–1962) ...nian ideal]], an ideal ''I'' in ''R'' for which the Krull dimension of the quotient ring ''R/I'' is 0 ...

...tion.<!-- this should be rephrased: When working with unfamiliar algebraic objects, one can use these to approximate with the more familiar. --> ...enerator of the category of abelian groups (since every abelian group is a quotient of a [[free abelian group]]). This is the origin of the term ''generator''. ...

...alled a [[coslice category]] denoted <math>\{ \bull \} /</math>'''Top'''.) Objects in this category are continuous maps <math>\{ \bull \} \to X.</math> Such m ...of the quotient is the image of the basepoint in <math>X</math> under the quotient map. ...

{{for|quotient spaces in linear algebra|quotient space (linear algebra)}} ...|thumb|Illustration of the construction of a [[topological sphere]] as the quotient space of a [[Disk (mathematics)|disk]], by ''gluing'' together to a single ...

{{about|quotients of vector spaces|quotients of topological spaces|Quotient space (topology)}} ...by "collapsing" <math>N</math> to zero. The space obtained is called a '''quotient space''' and is denoted <math>V/N</math> (read "<math>V</math> mod <math>N< ...

A '''quotient group''' or '''factor group''' is a [[math]]ematical [[group (mathematics)| ...txt|Dummit|Foote|2003|p=95}}</ref>) use this notation to refer only to the quotient group, with the appearance of this notation implying that <math>H</math> is ...

...not always possible in the smaller category of schemes, such as taking the quotient of a [[free action]] by a [[finite group]] (cf. the [[Keel–Mori theorem]]). ...paces over a field are schemes, though there are non quasi-separated group objects that are not schemes. ...

...sider [[quotient ring|quotients]] by [[free ring]]s to construct universal objects, C*-algebras must be realizable as algebras of bounded operators on a Hilbe ...l C*-relation'' on ''X'' is the category <math>\mathcal{F}_{X}</math> with objects consisting of pairs (''j'', ''A''), where ''A'' is a C*-algebra and ''j'' i ...

...ath> subject to special <math>+</math> and <math>\cdot</math> operations. (Quotient ring notation almost always uses a [[fraction slash]] "{{tmath|1= / }}"; st Quotient rings are distinct from the so-called "quotient field", or [[field of fractions]], of an [[integral domain]] as well as fro ...

...eserving maps called ''homomorphisms'' are vital in the study of algebraic objects. There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These construc ...

...rated by {{math|Σ}}. The monoid is then presented as the [[quotient monoid|quotient]] of the free monoid (or the free semigroup) by these relations. This is an ...ite) [[binary relation]] {{mvar|R}} on {{math|Σ^∗}}. To form the quotient monoid, these relations are extended to [[monoid congruence]]s as follows: ...

{{Short description|Category whose objects are topological spaces and whose morphisms are continuous maps}} [[compactly generated space|compactly generated spaces]] as objects and continuous maps as morphisms or with the [[category of compactly genera ...

...</sub>'' and the norm of an ideal ''I'' is given by the cardinality of the quotient ring ''O_K''/''I''. In this case, the appropriate generalisation ...ts, i.e. objects which cannot be decomposed as a direct product of nonzero objects. Some typical examples are the following. ...

* [[Factorization]], the decomposition of an object into a product of other objects * The act of forming a [[Quotient group|factor group]] or [[quotient ring]] in abstract algebra ...