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| In [[mathematics]] the '''additive identity''' of a [[Set (mathematics)|set]] which is equipped with the [[operation (mathematics)|operation]] of [[addition]] is an [[element (mathematics)|element]] which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the [[number]] [[0 (number)|0]] from [[elementary mathematics]], but additive identities occur in other mathematical structures where addition is defined, such as in [[group (mathematics)|groups]] and [[ring (mathematics)|rings]].
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| ==Elementary examples==
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| * The additive identity familiar from [[elementary mathematics]] is zero, denoted [[0 (number)|0]]. For example,
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| *: 5 + 0 = 5 = 0 + 5
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| * In the [[natural number]]s '''N''' and all of its [[superset]]s (the [[integer]]s '''Z''', the [[rational number]]s '''Q''', the [[real number]]s '''R''', or the [[complex number]]s '''C'''), the additive identity is 0. Thus for any one of these [[number]]s ''n'',
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| *: ''n'' + 0 = ''n'' = 0 + ''n''
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| ==Formal definition==
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| Let ''N'' be a [[Set (mathematics)|set]] which is closed under the [[operation (mathematics)|operation]] of [[addition]], denoted [[+]]. An additive identity for ''N'' is any element ''e'' such that for any element ''n'' in ''N'',
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| : ''e'' + ''n'' = ''n'' = ''n'' + ''e''
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| Example: The formula is n + 0 = n = 0 + n.
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| ==Further examples==
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| * In a [[group (mathematics)|group]] the additive identity is the [[identity element]] of the group, is often denoted 0, and is unique (see below for proof).
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| * A [[ring (mathematics)|ring]] or [[field (mathematics)|field]] is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the [[multiplicative identity]] [[1 (number)|1]] if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is [[trivial (mathematics)|trivial]] (proved below).
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| * In the ring M<sub>''m''×''n''</sub>(''R'') of ''m'' by ''n'' [[matrix (mathematics)|matrices]] over a ring ''R'', the additive identity is denoted '''0''' and is the ''m'' by ''n'' matrix whose entries consist entirely of the identity element 0 in ''R''. For example, in the 2 by 2 matrices over the integers M<sub>2</sub>('''Z''') the additive identity is
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| *:<math>0 = \begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}</math>
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| *In the [[quaternions]], 0 is the additive identity.
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| *In the ring of [[function (mathematics)|function]]s from '''R''' to '''R''', the function [[map (mathematics)|mapping]] every number to 0 is the additive identity.
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| *In the [[Abelian group|additive group]] of [[vector (geometric)|vector]]s in '''R'''<sup>''n''</sup>, the origin or [[zero vector]] is the additive identity.
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| ==Proofs==
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| ===The additive identity is unique in a group===
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| Let (''G'', +) be a group and let 0 and 0' in ''G'' both denote additive identities, so for any ''g'' in ''G'',
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| : 0 + ''g'' = ''g'' = ''g'' + 0 and 0' + ''g'' = ''g'' = ''g'' + 0' | |
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| It follows from the above that
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| : (0') = (0') + 0 = 0' + (0) = (0)
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| ===The additive identity annihilates ring elements===
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| In a system with a multiplication operation that distributes over addition, the additive identity is a multiplicative [[absorbing element]], meaning that for any ''s'' in ''S'', ''s''·0 = 0. This can be seen because:
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| :<math>\begin{align}
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| s \cdot 0 &= s \cdot (0 + 0) = s \cdot 0 + s \cdot 0 \\
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| \Rightarrow s \cdot 0 &= s \cdot 0 - s \cdot 0 \\
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| \Rightarrow s \cdot 0 &= 0
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| \end{align}</math>
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| ===The additive and multiplicative identities are different in a non-trivial ring===
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| Let ''R'' be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, or 0 = 1. Let ''r'' be any [[element (mathematics)|element]] of ''R''. Then
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| : ''r'' = ''r'' × 1 = ''r'' × 0 = 0
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| proving that ''R'' is trivial, that is, ''R'' = {0}. The [[contrapositive]], that if ''R'' is non-trivial then 0 is not equal to 1, is therefore shown.
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| ==See also==
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| *[[0 (number)]]
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| *[[Additive inverse]]
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| *[[Identity element]]
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| *[[Multiplicative identity]]
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| ==References==
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| *David S. Dummit, Richard M. Foote, ''Abstract Algebra'', Wiley (3d ed.): 2003, ISBN 0-471-43334-9.
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| ==External links==
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| *{{PlanetMath | urlname=UniquenessOfAdditiveIdentityInARing2 | title=uniqueness of additive identity in a ring | id=5676}}
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| *{{MathWorld | urlname=AdditiveIdentity | title=Additive Identity | author=Margherita Barile}}
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| [[Category:Abstract algebra]]
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| [[Category:Elementary algebra]]
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| [[Category:Group theory]]
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| [[Category:Ring theory]]
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| [[Category:Zero]]
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The writer's title is Andera and she thinks it seems fairly good. Office supervising is my occupation. The preferred hobby for him and his kids is to play lacross and he'll be beginning something else along with it. I've usually loved living in Mississippi.
Feel free to surf to my webpage - phone psychic (http://cspl.postech.ac.kr/)