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| {{ other uses2|Zero}}
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| In [[mathematics]], a '''zero element''' is one of several generalizations of [[0 (number)|the number zero]] to other [[algebraic structure]]s. These alternate meanings may or may not reduce to the same thing, depending on the context.
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| ==Additive identities==
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| An '''[[additive identity]]''' is the [[identity element]] in an [[Abelian group|additive group]]. It generalises the property {{nowrap|1=0 + ''x'' = ''x''}}. Examples include:
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| *The '''[[zero vector]]''' under [[vector addition]]
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| *The '''zero function''' or '''zero map''' under function addition {{nowrap|1=(''f'' + ''g'')(''x'') = ''f''(''x'') + ''g''(''x'')}}, defined by {{nowrap|1=''z''(''x'') = 0}}, since {{nowrap|1=''z'' + ''f'' = ''f''}}
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| *The '''[[empty set]]''' under [[Union (set theory)|set union]]
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| *An '''[[empty sum]]''' or '''empty [[coproduct]]'''
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| *An '''[[Initial and terminal objects|initial object]]''' in a [[category (mathematics)|category]] (an empty coproduct, and so an identity under [[coproduct]]s)
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| ==Absorbing elements==
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| An '''[[absorbing element]]''' in a multiplicative [[semigroup]] or [[semiring]] generalises the property {{nowrap|1=0 × ''x'' = 0}}. Examples include:
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| *The '''[[empty set]]''', which is an absorbing element under [[Cartesian product]] of sets, since {{nowrap|1={ } × ''S'' = { }<nowiki/>}}
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| *The '''zero function''' or '''zero map''' under function multiplication {{nowrap|1=(''f'' × ''g'')(''x'') = ''f''(''x'') × ''g''(''x'')}}, defined by {{nowrap|1=''z''(''x'') = 0}}, since {{nowrap|1=''z'' × ''f'' = ''z''}}
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| Many absorbing elements are also additive identities, including the empty set and the zero function. Another important example is the distinguished element 0 in a '''[[field (mathematics)|field]]''' or '''[[ring (mathematics)|ring]]''', which is both the additive identity and the multiplicative absorbing element, and whose [[principal ideal]] is the smallest ideal.
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| ==Zero objects==
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| A '''zero object''' in a [[category (mathematics)|category]] is both an [[Initial and terminal objects|initial and terminal object]] (and so an identity under both [[coproduct]]s and [[Product (category theory)|products]]). For example, the trivial structure (containing only the identity) is a zero object in categories where morphisms must map identities to identities. Specific examples include:
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| *The '''[[trivial group]]''', containing only the identity (a zero object in the [[category of groups]])
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| *The '''zero module''', containing only the identity (a zero object in the category of [[module (mathematics)|modules]] over a ring)
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| ==Zero morphisms==
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| A '''[[zero morphism]]''' in a [[category (mathematics)|category]] is a generalised absorbing element under [[function composition]]: any morphism composed with a zero morphism gives a zero morphism. Specifically, if {{nowrap|0<sub>''XY''</sub> : ''X'' → ''Y''}} is the zero morphism among morphisms from ''X'' to ''Y'', and {{nowrap|''f'' : ''A'' → ''X''}} and {{nowrap|''g'' : ''Y'' → ''B''}} are arbitrary morphisms, then {{nowrap|''g'' ∘ 0<sub>''XY''</sub> {{=}} 0<sub>''XB''</sub>}} and {{nowrap|0<sub>''XY''</sub> ∘ ''f'' {{=}} 0<sub>''AY''</sub>}}.
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| If a category has a zero object '''0''', then there are canonical morphisms {{nowrap|''X'' → '''0'''}} and {{nowrap|'''0''' → ''Y'',}} and composing them gives a zero morphism {{nowrap|0<sub>''XY''</sub> : ''X'' → ''Y''}}. In the [[category of groups]], for example, zero morphisms are morphisms which always return group identities, thus generalising the function {{nowrap|''z''(''x'') {{=}} 0.}}
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| ==Least elements==
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| A '''[[least element]]''' in a [[partially ordered set]] or [[Lattice (order)|lattice]] may sometimes be called a zero element, and written either as 0 or ⊥.
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| ==Zero module==
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| In [[mathematics]], the '''zero module''' is the [[module (mathematics)|module]] consisting of only the additive [[identity element|identity]] for the module's [[addition]] function. In the [[integer]]s, this identity is [[0 (number)|zero]], which gives the name ''zero module''. That the zero module is in fact a module is simple to show; it is closed under addition and [[multiplication]] trivially.
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| ==Zero ideal==
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| In [[mathematics]], the '''zero ideal''' in a [[Ring (mathematics)|ring]] <math>R</math> is the ideal <math>\{ 0 \}</math> consisting of only the additive identity (or [[0 (number)|zero]] element). It is immediate to show that this is an [[ideal (ring theory)|ideal]].
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| ==Zero matrix==
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| In [[mathematics]], particularly [[linear algebra]], a '''zero matrix''' is a [[matrix (mathematics)|matrix]] with all its entries being [[0 (number)|zero]]. Some examples of zero matrices are
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| :<math>
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| 0_{1,1} = \begin{bmatrix}
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| 0 \end{bmatrix}
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| ,\
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| 0_{2,2} = \begin{bmatrix}
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| 0 & 0 \\
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| 0 & 0 \end{bmatrix}
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| ,\
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| 0_{2,3} = \begin{bmatrix}
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| 0 & 0 & 0 \\
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| 0 & 0 & 0 \end{bmatrix}
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| ,\
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| </math>
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| The set of ''m''×''n'' matrices with entries in a [[ring (mathematics)|ring]] K forms a module <math>K_{m,n} \,</math>. The zero matrix <math>0_{K_{m,n}} \, </math> in <math>K_{m,n} \, </math> is the matrix with all entries equal to <math>0_K \, </math>, where <math>0_K \, </math> is the additive identity in K.
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| :<math> | |
| 0_{K_{m,n}} = \begin{bmatrix}
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| 0_K & 0_K & \cdots & 0_K \\
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| 0_K & 0_K & \cdots & 0_K \\
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| \vdots & \vdots & & \vdots \\
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| 0_K & 0_K & \cdots & 0_K \end{bmatrix}
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| </math>
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| The zero matrix is the additive identity in <math>K_{m,n} \, </math>. That is, for all <math>A \in K_{m,n} \, </math> it satisfies
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| :<math>0_{K_{m,n}}+A = A + 0_{K_{m,n}} = A</math>
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| There is exactly one zero matrix of any given size ''m''×''n'' having entries in a given ring, so when the context is clear one often refers to ''the'' zero matrix. In general the zero element of a ring is unique and typically denoted as 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.
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| The zero matrix represents the [[linear transformation]] sending all vectors to the [[zero vector]].
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| ==Zero tensor==
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| In [[mathematics]], the '''zero tensor''' is a [[tensor]], of any order, all of whose components are [[0 (number)|zero]]. The zero tensor of order 1 is sometimes known as the [[zero vector]].
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| Taking a [[tensor product]] of any tensor with any zero tensor results in another zero tensor. Adding the zero tensor is equivalent to the identity operation.
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| ==Zero divisor==
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| A '''[[zero divisor]]''' in a [[ring (mathematics)|ring]] ''R'' is a non-zero element ''a'' ∈ ''R'' such that ''ab'' = 0 for some non-zero ''b'' ∈ ''R''.
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| ==Zerosumfree monoid==
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| In [[abstract algebra]], an additive [[monoid]] <math>(M, 0, +)</math> is said to be '''zerosumfree''' if nonzero elements do not sum to zero. Formally:
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| :<math>(\forall a,b\in M)\ a + b = 0 \implies a = 0 = b \!</math>
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| This means that the only way zero can be expressed as a sum is as <math>0 + 0</math>.
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| ==See also==
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| *[[Zero object]]
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| *[[Zero of a function]]
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| *[[Zero]] non mathematical uses.
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| [[Category:Zero]]
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The author's title is Christy. I am presently a journey agent. My wife and I reside in Mississippi but now I'm considering other options. To perform lacross is one of the things she loves most.
Stop by my weblog ... phone psychic