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{{about|zero objects or trivial objects in algebraic structures|zero object in a category|Initial and terminal objects|trivial representation|trivial representation}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
<!-- {{redirect|{0}|0 (disambiguation)}} -->
{{Merge from|trivial group|discuss=Talk:Zero object (algebra)#Zero object (algebra)|date=February 2012}}
{{refimprove|date=February 2012}}
[[Image:Terminal and initial object.svg|thumb|right|[[Morphism]]s to and from the zero object]]
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 (mathematics)|set]] it is a [[singleton (mathematics)|singleton]], and also has a [[trivial group|trivial]] structure of [[abelian group]]. Aforementioned group structure usually identified as the [[addition]], and the only element is called [[zero]]&nbsp;0, so the object itself is denoted as {{math|{0}{{void}}}}. One often refers to ''the'' trivial object (of a specified [[category (mathematics)|category]]) since every trivial object is [[isomorphism|isomorphic]] to any other (under a unique isomorphism).  


Instances of the zero object include, but are not limited to the following:
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* As a [[group (mathematics)|group]], the '''trivial group'''.
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* As a [[ring (mathematics)|ring]], the '''trivial ring'''.
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* As a [[module (mathematics)|module]] (over a [[ring (algebra)|ring]]&nbsp;{{mvar|R}}), the '''zero module'''. The term ''trivial module'' is also used, although it is ambiguous.
* As a [[vector space]] (over a [[field (mathematics)|field]]&nbsp;{{mvar|R}}), the '''zero vector space''', '''zero-dimensional vector space'''  or just '''zero space'''; see [[#Vector space|below]].
* As an [[algebra over a field]] or [[algebra over a ring]], the '''trivial algebra'''.
These objects are described jointly not only based on the common singleton and trivial group structure, but also because of [[#Properties|shared category-theoretical properties]].


{{anchor|Module}}In the last three cases the [[scalar multiplication]] by an element of the base ring (or field) is defined as:
Registered users will be able to choose between the following three rendering modes:  
: {{math|1= κ0 = 0 }}, where {{math|κ ∈ ''R''}}.
The most general of them, the zero module, is a [[finitely-generated module]] with an [[empty set|empty]] generating set.


{{anchor|Algebra}}{{anchor|Ring}}For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, {{math|1=0 × 0 = 0}}, because there are no non-zero elements. This structure is [[associativity|associative]] and [[commutative]]. A ring {{mvar|R}} which has both an additive and multiplicative identity is trivial if and only if {{nowrap|1=1 = 0}}, since this equality implies that for all {{mvar|r}} within {{mvar|R}},
'''MathML'''
:<math>r = r \times 1 = r \times 0 = 0. \,</math>
:<math forcemathmode="mathml">E=mc^2</math>
In this case it is possible to define [[division by zero]], since the single element is its own multiplicative inverse. Some properties of {0} depend on exact definition of the multiplicative identity; see the section [[#Unital structures|Unital structures]] below.


Any '''trivial algebra''' is also a trivial ring. A trivial [[algebra over a field]] is simultaneously a zero vector space considered [[#Vector space|below]]. Over a [[commutative ring]], a trivial [[algebra over a ring|algebra]] is simultaneously a zero module.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


The trivial ring is an example of a [[zero ring]]. Likewise, a trivial algebra is an example of a [[Algebra over a field#Zero algebras|zero algebra]].
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


; {{visible anchor|Vector space}}
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
The zero-dimensional vector space is an especially ubiquitous example of a zero object, a [[vector space]] over a field with an empty [[basis (linear algebra)|basis]]. It therefore has [[dimension (mathematics)|dimension]] zero. It is also a trivial group over [[addition]], and a ''trivial module'' [[#Module|mentioned above]].


== Properties ==<!-- linked from the lede -->
==Demos==
{| table align=right valign=center width="32em" style="margin-left:2em"
|- align=center
| bgcolor=#66FFFF align=right | 2<span style="font-size:160%">↕</span>&nbsp;
| <math>\begin{bmatrix}0 \\ 0\end{bmatrix}</math>
| style="font-size:200%" | =
| <math>\begin{bmatrix} \,\\ \,\end{bmatrix}</math>
| <span style="font-size:40%; font-weight:900">[</span> <span style="font-size:40%; font-weight:900">]</span>
| bgcolor=#66FFFF align=left | &nbsp;‹0
|- bgcolor=#66FFFF align=center
|
| ↔<br/>1
|
| ^<br/>0
| ↔<br/>1
|
|-
| colspan=6 style="font-size:75%" | Element of the zero space, written as empty [[column vector]] (rightmost one), <br/> is multiplied by 2×0 [[empty matrix]] to obtain 2-dimensional zero vector </br> (leftmost). Rules of [[matrix multiplication]] are respected.
|}
The trivial ring, zero module and zero vector space are [[zero object]]s of the corresponding [[category (mathematics)|categories]], namely <span class="nounderlines">[[Category of pseudo-rings|'''Rng''']], [[Category of modules|{{mvar|R}}-'''Mod''']] and [[Category of vector spaces|'''Vect'''<sub>{{mvar|R}}</sub>]]</span>.


The zero object, by definition, must be a terminal object, which means that a [[morphism]]&nbsp;{{math|''A'' → {0}{{void}}}} must exist and be unique for an arbitrary object&nbsp;{{mvar|A}}. This morphism maps any element of&nbsp;{{mvar|A}} to&nbsp;{{math|0}}.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


The zero object, also by definition, must be an initial object, which means that a morphism&nbsp;{{math|{0} → ''A''}} must exist and be unique for an arbitrary object&nbsp;{{mvar|A}}. This morphism maps {{math|0}}, the only element of&nbsp;{{math|{0}{{void}}}}, to the zero element&nbsp;{{math|0 ∈ ''A''}}, called the [[zero vector]] in vector spaces. This map is a [[monomorphism]], and hence its image is isomorphic to&nbsp;{0}. For modules and vector spaces, this [[subset]]&nbsp;{{math|{0} ⊂ ''A''}} is the only empty-generated [[submodule]] (or 0-dimensional [[linear subspace]]) in each module (or vector space)&nbsp;{{mvar|A}}.


=== Unital structures ===<!-- linked from the lede -->
* accessibility:
The {0} object is a [[terminal object]] of any algebraic structure where it exists, like it was described for examples above. But its existence and, if it exists, the property to be an [[initial object]] (and hence, a ''zero object'' in the [[category theory|category-theoretical]] sense) depend on exact definition of the [[multiplicative identity]]&nbsp;1 in a specified structure.
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


If the definition of&nbsp;1 requires that {{math|1 ≠ 0}}, then the {0} object cannot exist because it may contain only one element. In particular, the zero ring is not a [[field (mathematics)|field]]. If mathematicians sometimes talk about a [[field with one element]], this abstract and somewhat mysterious mathematical object is not a field.
==Test pages ==


In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the {0} object can exist. But not as initial object because identity-preserving morphisms from {0} to any object where {{math|1 ≠ 0}} do not exist. For example, in the [[category of rings]] '''Ring''' the ring of [[integer]]s&nbsp;'''Z''' is the initial object, not&nbsp;{0}.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


If an algebraic structure requires the multiplicative identity, but does not require neither its preserving by morphisms nor {{math|1 ≠ 0}}, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
== Notation ==
==Bug reporting==
Zero vector spaces and zero modules are usually denoted by 0 (instead of {0}). This is always the case when they occur in an [[exact sequence]].
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
== See also ==
* [[Triviality (mathematics)]]
* [[Examples of vector spaces]]
* [[Field with one element]]
* [[Zero element (disambiguation)]]
* [[List of zero terms]]
 
== External links ==
* {{cite book | author=David Sharpe | title=Rings and factorization | publisher=[[Cambridge University Press]] | year=1987 | isbn=0-521-33718-6 | page=10 }}
* {{MathWorld|title=Trivial Module|id=TrivialModule|author=[[Margherita Barile|Barile, Margherita]]}}
* {{MathWorld|title=Zero Module|id=ZeroModule|author=Barile, Margherita}}
 
[[Category:Ring theory|0]]
[[Category:Linear algebra|0]]
[[Category:Zero|Object]]
[[Category:Objects (category theory)|0]]
 
<!-- these interwiki are actually of [[trivial ring]] -->
[[ca:Anell trivial]]
[[de:Nullring]]
[[es:Anillo trivial]]
[[fr:Anneau nul]]
[[nl:Triviale ring]]
[[pl:Pierścień trywialny]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

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Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

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