Q-gamma function - Revision history

en>Rjwilmsi: /* References */Added 2 dois to journal cites using AWB (10210)

2014-05-24T10:17:06Z

References: Added 2 dois to journal cites using AWB (10210)

← Older revision		Revision as of 12:17, 24 May 2014
Line 1:		Line 1:
	~~{{about\|zero objects or trivial objects in algebraic structures\|zero object in a category\|Initial and terminal objects\|trivial representation\|Trivial representation}}~~		The writer is called Irwin. In her expert life she is a payroll clerk but she's always needed her personal business. What I love doing is to collect badges but I've been using on new issues recently. Puerto Rico is exactly where he's usually been living but she requirements to move simply because of her family members.<br><br>Also visit my web-site [http://c-batang.Co.kr/?document_srl=485588&mid=levelup_board home std test kit]
	~~<!-- {{redirect\|{0}\|0 (disambiguation)}} -->~~
	~~{{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]] 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:~~
	* As a [[group (mathematics)\|group]], the '''trivial group'''.
	* As a [[ring (mathematics)\|ring]], the '''trivial ring'''.
	* As a [[module (mathematics)\|module]] (over a [[ring (algebra)\|ring]] {{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]] {{mvar\|R}}), the ''~~'zero vector space''', '''zero-dimensional vector space''' or just '''zero space'''.~~
	* 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:~~
	~~: {{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}},
	~~:<math>r = r \times 1 = r \times 0 = 0. \,</math>~~
	~~In this case it is possible to define [[division by zero]], since the single element is its own multiplicative inverse. Some properties~~ of ~~{{math\|{0}{{void}}}} 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.~~

	~~The trivial ring is an example of a [[pseudo-ring#Pseudo-rings_of_square_zero\|pseudo-ring of square zero]]. A trivial algebra is an example of a [[Algebra over a field#Zero algebras\|zero algebra]].~~

	The zero-dimensional {{visible anchor\|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 -->~~
	~~{\| table align=right valign=center width="32em" style="margin-left:2em"~~
	~~\|- align=center~~
	~~\| bgcolor=#66FFFF align=right \| 2<span style="font-size:160%">↕</span> ~~
	~~\| <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 \|  ‹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]] {{math\|''A'' → {0}{{void}}}} must exist and be unique for an arbitrary object {{mvar\|A}}. This morphism maps any element of {{mvar\|A}} to {{math\|0}}.

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

	~~=== Unital structures ===<!-- linked from the lede -->~~
	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]] 1 in a specified structure.

	If the definition of 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.

	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 '''Z''' is the initial object, not {0}.

	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.

	~~== Notation ==~~
	~~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]].~~

	~~== See also ==~~
	* [[Triviality (mathematics)]]
	* [[Examples of vector spaces]]
	* [[Field with one element]]
	* [[Empty semigroup]]
	* [[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=[http://~~books~~.~~google~~.fr/~~books~~?id=~~Nmg4AAAAIAAJ~~&pg=~~PA10 10] : ''trivial ring''}}~~
	* {{MathWorld\|title=Trivial Module\|id=TrivialModule\|author=[[Margherita Barile\|Barile, Margherita]]}}
	* {{MathWorld\|title=Zero Module\|id=ZeroModule\|author=Barile, Margherita}}

	~~<!-- these interwiki are actually of [[trivial ring]] -->~~

	~~[[Category:Ring theory\|0]]~~
	~~[[Category:Linear algebra\|0]]~~
	~~[[Category:Zero\|Object]]~~
	~~[[Category:Objects (category theory)\|0]~~]

en>Tabletop: Spell funtion => function (15)

2013-10-04T14:47:31Z

Spell funtion => function (15)

← Older revision		Revision as of 16:47, 4 October 2013
Line 1:		Line 1:
	~~The author~~ is called ~~Irwin Wunder~~ but it's not the most ~~masucline title out~~ there. ~~Supervising~~ is ~~my profession~~. ~~His family members life in South Dakota but his spouse desires them~~ to ~~transfer~~. ~~To gather cash~~ is a ~~thing that I~~'~~m totally addicted to~~.<br><br>~~My webpage~~ :: ~~std testing at home (~~[~~http~~://~~www~~.~~egitimpark~~.~~net~~/~~siir~~/~~groups/clear~~-up-a-~~candida~~-with-these-~~tips/ visit the following internet page~~])		{{about\|zero objects or trivial objects in algebraic structures\|zero object in a category\|Initial and terminal objects\|trivial representation\|Trivial representation}}
			<!-- {{redirect\|{0}\|0 (disambiguation)}} -->
			{{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]] 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:
			* As a [[group (mathematics)\|group]], the '''trivial group'''.
			* As a [[ring (mathematics)\|ring]], the '''trivial ring'''.
			* As a [[module (mathematics)\|module]] (over a [[ring (algebra)\|ring]] {{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]] {{mvar\|R}}), the '''zero vector space''', '''zero-dimensional vector space''' or just '''zero space'''.
			* 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:
			: {{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}},
			:<math>r = r \times 1 = r \times 0 = 0. \,</math>
			In this case it is possible to define [[division by zero]], since the single element is its own multiplicative inverse. Some properties of {{math\|{0}{{void}}}} 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.

			The trivial ring is an example of a [[pseudo-ring#Pseudo-rings_of_square_zero\|pseudo-ring of square zero]]. A trivial algebra is an example of a [[Algebra over a field#Zero algebras\|zero algebra]].

			The zero-dimensional {{visible anchor\|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 -->
			{\| table align=right valign=center width="32em" style="margin-left:2em"
			\|- align=center
			\| bgcolor=#66FFFF align=right \| 2<span style="font-size:160%">↕</span>
			\| <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 \|  ‹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]] {{math\|''A'' → {0}{{void}}}} must exist and be unique for an arbitrary object {{mvar\|A}}. This morphism maps any element of {{mvar\|A}} to {{math\|0}}.

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

			=== Unital structures ===<!-- linked from the lede -->
			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]] 1 in a specified structure.

			If the definition of 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.

			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 '''Z''' is the initial object, not {0}.

			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.

			== Notation ==
			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]].

			== See also ==
			* [[Triviality (mathematics)]]
			* [[Examples of vector spaces]]
			* [[Field with one element]]
			* [[Empty semigroup]]
			* [[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=[http://books.google.fr/books?id=Nmg4AAAAIAAJ&pg=PA10 10] : ''trivial ring''}}
			* {{MathWorld\|title=Trivial Module\|id=TrivialModule\|author=[[Margherita Barile\|Barile, Margherita]]}}
			* {{MathWorld\|title=Zero Module\|id=ZeroModule\|author=Barile, Margherita}}

			<!-- these interwiki are actually of [[trivial ring]] -->

			[[Category:Ring theory\|0]]
			[[Category:Linear algebra\|0]]
			[[Category:Zero\|Object]]
			[[Category:Objects (category theory)\|0]]

76.120.181.41: fix

2012-09-01T22:59:50Z

fix

New page

The author is called Irwin Wunder but it's not the most masucline title out there. Supervising is my profession. His family members life in South Dakota but his spouse desires them to transfer. To gather cash is a thing that I'm totally addicted to.<br><br>My webpage :: std testing at home ([http://www.egitimpark.net/siir/groups/clear-up-a-candida-with-these-tips/ visit the following internet page])

← Older revision		Revision as of 12:17, 24 May 2014
Line 1:		Line 1:
	~~{{about\|zero objects or trivial objects in algebraic structures\|zero object in a category\|Initial and terminal objects\|trivial representation\|Trivial representation}}~~		The writer is called Irwin. In her expert life she is a payroll clerk but she's always needed her personal business. What I love doing is to collect badges but I've been using on new issues recently. Puerto Rico is exactly where he's usually been living but she requirements to move simply because of her family members.<br><br>Also visit my web-site [http://c-batang.Co.kr/?document_srl=485588&mid=levelup_board home std test kit]
	~~<!-- {{redirect\|{0}\|0 (disambiguation)}} -->~~
	~~{{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]] 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:~~
	* As a [[group (mathematics)\|group]], the '''trivial group'''.
	* As a [[ring (mathematics)\|ring]], the '''trivial ring'''.
	* As a [[module (mathematics)\|module]] (over a [[ring (algebra)\|ring]] {{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]] {{mvar\|R}}), the ''~~'zero vector space''', '''zero-dimensional vector space''' or just '''zero space'''.~~
	* 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:~~
	~~: {{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}},
	~~:<math>r = r \times 1 = r \times 0 = 0. \,</math>~~
	~~In this case it is possible to define [[division by zero]], since the single element is its own multiplicative inverse. Some properties~~ of ~~{{math\|{0}{{void}}}} 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.~~

	~~The trivial ring is an example of a [[pseudo-ring#Pseudo-rings_of_square_zero\|pseudo-ring of square zero]]. A trivial algebra is an example of a [[Algebra over a field#Zero algebras\|zero algebra]].~~

	The zero-dimensional {{visible anchor\|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 -->~~
	~~{\| table align=right valign=center width="32em" style="margin-left:2em"~~
	~~\|- align=center~~
	~~\| bgcolor=#66FFFF align=right \| 2<span style="font-size:160%">↕</span> ~~
	~~\| <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 \|  ‹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]] {{math\|''A'' → {0}{{void}}}} must exist and be unique for an arbitrary object {{mvar\|A}}. This morphism maps any element of {{mvar\|A}} to {{math\|0}}.

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

	~~=== Unital structures ===<!-- linked from the lede -->~~
	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]] 1 in a specified structure.

	If the definition of 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.

	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 '''Z''' is the initial object, not {0}.

	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.

	~~== Notation ==~~
	~~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]].~~

	~~== See also ==~~
	* [[Triviality (mathematics)]]
	* [[Examples of vector spaces]]
	* [[Field with one element]]
	* [[Empty semigroup]]
	* [[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=[http://~~books~~.~~google~~.fr/~~books~~?id=~~Nmg4AAAAIAAJ~~&pg=~~PA10 10] : ''trivial ring''}}~~
	* {{MathWorld\|title=Trivial Module\|id=TrivialModule\|author=[[Margherita Barile\|Barile, Margherita]]}}
	* {{MathWorld\|title=Zero Module\|id=ZeroModule\|author=Barile, Margherita}}

	~~<!-- these interwiki are actually of [[trivial ring]] -->~~

	~~[[Category:Ring theory\|0]]~~
	~~[[Category:Linear algebra\|0]]~~
	~~[[Category:Zero\|Object]]~~
	~~[[Category:Objects (category theory)\|0]~~]

← Older revision		Revision as of 16:47, 4 October 2013
Line 1:		Line 1:
	~~The author~~ is called ~~Irwin Wunder~~ but it's not the most ~~masucline title out~~ there. ~~Supervising~~ is ~~my profession~~. ~~His family members life in South Dakota but his spouse desires them~~ to ~~transfer~~. ~~To gather cash~~ is a ~~thing that I~~'~~m totally addicted to~~.<br><br>~~My webpage~~ :: ~~std testing at home (~~[~~http~~://~~www~~.~~egitimpark~~.~~net~~/~~siir~~/~~groups/clear~~-up-a-~~candida~~-with-these-~~tips/ visit the following internet page~~])		{{about\|zero objects or trivial objects in algebraic structures\|zero object in a category\|Initial and terminal objects\|trivial representation\|Trivial representation}}
			<!-- {{redirect\|{0}\|0 (disambiguation)}} -->
			{{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]] 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:
			* As a [[group (mathematics)\|group]], the '''trivial group'''.
			* As a [[ring (mathematics)\|ring]], the '''trivial ring'''.
			* As a [[module (mathematics)\|module]] (over a [[ring (algebra)\|ring]] {{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]] {{mvar\|R}}), the '''zero vector space''', '''zero-dimensional vector space''' or just '''zero space'''.
			* 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:
			: {{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}},
			:<math>r = r \times 1 = r \times 0 = 0. \,</math>
			In this case it is possible to define [[division by zero]], since the single element is its own multiplicative inverse. Some properties of {{math\|{0}{{void}}}} 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.

			The trivial ring is an example of a [[pseudo-ring#Pseudo-rings_of_square_zero\|pseudo-ring of square zero]]. A trivial algebra is an example of a [[Algebra over a field#Zero algebras\|zero algebra]].

			The zero-dimensional {{visible anchor\|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 -->
			{\| table align=right valign=center width="32em" style="margin-left:2em"
			\|- align=center
			\| bgcolor=#66FFFF align=right \| 2<span style="font-size:160%">↕</span>
			\| <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 \|  ‹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]] {{math\|''A'' → {0}{{void}}}} must exist and be unique for an arbitrary object {{mvar\|A}}. This morphism maps any element of {{mvar\|A}} to {{math\|0}}.

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

			=== Unital structures ===<!-- linked from the lede -->
			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]] 1 in a specified structure.

			If the definition of 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.

			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 '''Z''' is the initial object, not {0}.

			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.

			== Notation ==
			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]].

			== See also ==
			* [[Triviality (mathematics)]]
			* [[Examples of vector spaces]]
			* [[Field with one element]]
			* [[Empty semigroup]]
			* [[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=[http://books.google.fr/books?id=Nmg4AAAAIAAJ&pg=PA10 10] : ''trivial ring''}}
			* {{MathWorld\|title=Trivial Module\|id=TrivialModule\|author=[[Margherita Barile\|Barile, Margherita]]}}
			* {{MathWorld\|title=Zero Module\|id=ZeroModule\|author=Barile, Margherita}}

			<!-- these interwiki are actually of [[trivial ring]] -->

			[[Category:Ring theory\|0]]
			[[Category:Linear algebra\|0]]
			[[Category:Zero\|Object]]
			[[Category:Objects (category theory)\|0]]