- Additive identity
In

mathematics the**additive identity**of a set which is equipped with the operation ofaddition is an element which, when added to any element "x" in the set, yields "x". One of the most familiar additive identities is thenumber 0 fromelementary mathematics , but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.**Elementary examples*** The additive identity familiar from

elementary mathematics is zero, denoted 0. For example,

*: 5 + 0 = 5 = 0 + 5.

* In thenatural number s**N**and all of itssuperset s (theinteger s**Z**, therational number s**Q**, thereal number s**R**, or thecomplex number s**C**), the additive identity is 0. Thus for any one of thesenumber s "n",

*: "n" + 0 = "n" = 0 + "n".**Formal definition**Let "N" be a set which is closed under the operation of

addition , denoted+ . An additive identity for "N" is any element "e" such that for any element "n" in "N",: "e" + "n" = "n" = "n" + "e".**Further examples*** In a group the additive identity is the

identity element of the group, is often denoted 0, and is unique (see below for proof).

* A ring or 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 themultiplicative identity 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 (proved below).

* In the ring M_{"m"×"n"}("R") of "m" by "n" 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_{2}(**Z**) the additive identity is

*:$0\; =\; egin\{pmatrix\}0\; 0\; \backslash \; 0\; 0end\{pmatrix\}.$

*In thequaternions , 0 is the additive identity.

*In the ring of functions from**R**to**R**, the function mapping every number to 0 is the additive identity.

*In theadditive group of vectors in**R**^{"n"}, the origin orzero vector is the additive identity.**Proofs****The additive identity is unique in a group**Let ("G", +) be a group and let 0 and 0' in "G" both denote additive identities, so for any "g" in "G",: 0 + "g" = "g" = "g" + 0 and 0' + "g" = "g" = "g" + 0'.

It follows from the above that: 0 + (0') = (0') = (0') + 0 and 0' + (0) = (0) = (0) + 0'which shows that 0 = 0'.

**The additive and multiplicative identities are different in a non-trivial ring**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 of "R". Then

: "r" = "r" × 1 = "r" × 0 = 0,

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.**The additive identity annihilates ring elements**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 "s"·0 = "s"·(0 + 0) = "s"·0 + "s"·0, so that, by cancellation "s"·0 = 0. Any number times 0 equals 0.**ee also***

0

*Additive inverse

*Identity element

*Multiplicative identity **References***David S. Dummit, Richard M. Foote, "Abstract Algebra", Wiley (3d ed.): 2003, ISBN 0-471-43334-9.

**External links***PlanetMath | urlname=UniquenessOfAdditiveIdentityInARing2 | title=uniqueness of additive identity in a ring | id=5676

*MathWorld | urlname=AdditiveIdentity | title=Additive Identity | author=Margherita Barile

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