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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}}
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| <!-- {{redirect|{0}|0 (disambiguation)}} -->
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| {{refimprove|date=February 2012}}
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| [[Image:Terminal and initial object.svg|thumb|right|[[Morphism]]s to and from the zero object]]
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| 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).
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| Instances of the zero object include, but are not limited to the following:
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| * 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]] {{mvar|R}}), the '''zero module'''. The term ''trivial module'' is also used, although it is ambiguous.
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| * As a [[vector space]] (over a [[field (mathematics)|field]] {{mvar|R}}), the '''zero vector space''', '''zero-dimensional vector space''' or just '''zero space'''.
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| * As an [[algebra over a field]] or [[algebra over a ring]], the '''trivial algebra'''.
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| 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]].
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| {{anchor|Module}}In the last three cases the [[scalar multiplication]] by an element of the base ring (or field) is defined as:
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| : {{math|1= κ0 = 0 }}, where {{math|κ ∈ ''R''}}.
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| The most general of them, the zero module, is a [[finitely-generated module]] with an [[empty set|empty]] generating set.
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| {{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}},
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| :<math>r = r \times 1 = r \times 0 = 0. \,</math>
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| 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.
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| 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.
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| 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]].
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| 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]].
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| == Properties ==<!-- linked from the lede -->
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| {| table align=right valign=center width="32em" style="margin-left:2em"
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| |- align=center
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| | bgcolor=#66FFFF align=right | 2<span style="font-size:160%">↕</span>
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| | <math>\begin{bmatrix}0 \\ 0\end{bmatrix}</math>
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| | style="font-size:200%" | =
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| | <math>\begin{bmatrix} \,\\ \,\end{bmatrix}</math>
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| | <span style="font-size:40%; font-weight:900">[</span> <span style="font-size:40%; font-weight:900">]</span>
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| | bgcolor=#66FFFF align=left | ‹0
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| |- bgcolor=#66FFFF align=center
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| | ↔<br/>1
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| | ^<br/>0
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| | ↔<br/>1
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| |-
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| | 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.
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| |}
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| 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>.
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| 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}}.
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| 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}}.
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| === Unital structures ===<!-- linked from the lede -->
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| 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.
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| 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.
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| 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}.
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| 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.
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| == Notation ==
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| 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]].
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| == See also ==
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| * [[Triviality (mathematics)]]
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| * [[Examples of vector spaces]]
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| * [[Field with one element]]
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| * [[Empty semigroup]]
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| * [[Zero element (disambiguation)]]
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| * [[List of zero terms]]
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| == External links ==
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| * {{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''}}
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| * {{MathWorld|title=Trivial Module|id=TrivialModule|author=[[Margherita Barile|Barile, Margherita]]}}
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| * {{MathWorld|title=Zero Module|id=ZeroModule|author=Barile, Margherita}}
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| <!-- these interwiki are actually of [[trivial ring]] -->
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| [[Category:Ring theory|0]]
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| [[Category:Linear algebra|0]]
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| [[Category:Zero|Object]]
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| [[Category:Objects (category theory)|0]]
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