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

*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 ...

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 ...

...except for an equivalent structure}}Two [[Mathematical object|mathematical objects]] {{mvar|a}} and {{mvar|b}} are called "equal '''up to''' an [[equivalence For example, "{{mvar|x}} is unique up to {{mvar|R}}" means that all objects {{mvar|x}} under consideration are in the same equivalence class with respe ...

...ery congruence relation has a corresponding [[Quotient (universal algebra)|quotient]] structure, whose elements are the [[equivalence class]]es (or '''congruen ...tegers, and arithmetic modulo <math>n</math> occurs on the corresponding [[quotient ring]]. ...

...m''', or just the '''homomorphism theorem''', relates the structure of two objects between which a [[homomorphism]] is given, and of the [[Kernel (algebra)|ke ...homomorphism <math>G \rarr G / N</math> (where <math>G / N</math> is the [[quotient group]] of <math>G</math> by <math>N</math>). If <math>N</math> is a [[subs ...

{{short description|Smallest normal subgroup by which the quotient is commutative}} ...is the [[Universal property|smallest]] [[normal subgroup]] such that the [[quotient group]] of the original group by this subgroup is [[abelian group|abelian]] ...

...athematics)|image]] of a homomorphism is [[isomorphism|isomorphic]] to the quotient by the kernel.<ref name="McKenzie Kernel"/><ref name="Dummit Group FIT" /> ...and further it is a [[normal subgroup]]. Thus, there is a corresponding [[quotient group]] <math>G/\ker{f}</math>. This is isomorphic to <math>f(G)</math>, th ...

...me constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, th ...ught more abstractly as [[Initial and terminal objects|initial or terminal objects]] of a [[comma category]] (see {{slink||Connection with comma categories}}, ...

...phism theorems''') are [[theorem]]s that describe the relationship among [[Quotient (universal algebra)|quotients]], [[homomorphism]]s, and [[subobject]]s. Ve # The image of <math>f</math> is [[group isomorphism|isomorphic]] to the [[quotient group]] <math>G / \ker f</math>. ...

...hism]]. That is, an arrow {{math|''f'' : ''X'' → ''Y''}} such that for all objects {{math|''Z''}} and all morphisms {{math|''g''₁, ''g''<sub>2</sub ...r all morphisms {{nowrap|''h'' : ''Z'' → ''X''}}, is [[injective]] for all objects ''Z''. ...

...hich is divided by the '''''divisor''''', and the result is called the ''[[quotient]]''. ...part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. F ...

.../C</math>, where the forward slash denotes the [[quotient space (topology)|quotient space]].<ref name=rotman>[[Joseph J. Rotman]], ''An Introduction to Algebra ...where <math>A \vee B</math> is the [[wedge sum]], and again, / denotes the quotient space.<ref name=rotman/> ...

...rnel (algebra)|ordinary kernel from group theory]] and the cokernel is the quotient map onto the [[cokernel|ordinary cokernel from group theory]]. ...gory of [[Set (mathematics)|sets]], where images and coimages exist, their objects are [[isomorphic]]. ...

...is [[Cancellation property|right-cancellative]] in the sense that, for all objects ''Z'' and all morphisms {{math|''g''₁, ''g''₂: ''Y'' ...R}} is surjective, we compose it with both the canonical [[quotient module|quotient map]] {{math|''g''₁ : ''Y'' → ''Y''/''f''(''X'')}} and the [[zer ...

...onfigurations of [[axiom]]s. [[Universal algebra]] abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by ...]. These equations induce [[equivalence class]]es on the free algebra; the quotient algebra then has the algebraic structure of a group. ...

...hematics)|category]] in which [[morphism]]s and [[Object (category theory)|objects]] can be added and in which [[Kernel (category theory)|kernel]]s and [[coke ...category is ''[[Additive category|additive]]'' if every [[finite set]] of objects has a [[biproduct]]. This means that we can form finite [[direct sum of mod ...

{{Short description|Mathematical concept for comparing objects}} ...<math>X</math>. It is also called the [[Equivalence relation#Quotient set|quotient set]] of <math>X</math> by <math>R</math>. ...

Groupoids are often used to reason about [[geometrical]] objects such as [[manifold]]s. {{harvs|txt|first=Heinrich|last=Brandt|authorlink=He ...More explicitly, a groupoid <math>G</math> is a set <math>G_0</math> of ''objects'' with ...

* [[Quotient group]] ==Mathematical objects making use of a group operation== ...