In

_{''m'' × ''n''}(''R'') of ''m'' by ''n'' _{2}(Z) the additive identity is
*:$0\; =\; \backslash begin0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\backslash end$
*In the ^{''n''}, the origin or

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, the additive identity of a set that is equipped with the operation of addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

is an element
Element may refer to:
Science
* Chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting only of atoms that all ...

which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

from elementary mathematics
300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children.
Elementary mathematics consists of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

, but additive identities occur in other mathematical structures where addition is defined, such as in groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic identi ...

and rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery)
A ring is a round band, usually of metal
A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...

.
Elementary examples

* The additive identity familiar fromelementary mathematics
300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children.
Elementary mathematics consists of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

is zero, denoted . For example,
*:$5\; +\; 0\; =\; 5\; =\; 0\; +\; 5.$
* In the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s N (if 0 is included), the integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s Z, the rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s Q, the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s R, and the complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s C, the additive identity is 0. This says that for a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

''n'' belonging to any of these sets,
*:$n\; +\; 0\; =\; n\; =\; 0\; +\; n.$
Formal definition

Let ''N'' be agroup
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

that is closed under the operation of addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

, denoted . An additive identity for ''N'', denoted ''e'', is an element in ''N'' such that for any element ''n'' in ''N'',
: ''e'' + ''n'' = ''n'' = ''n'' + ''e''.
Further examples

* In agroup
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

, the additive identity is the identity element
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of the group, is often denoted 0, and is unique (see below for proof).
* A ring or field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

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
In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...

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 Mmatrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

over a ring ''R'', the additive identity is the zero matrix, denoted O or 0, and is the ''m'' by ''n'' matrix whose entries consist entirely of the identity element 0 in ''R''. For example, in the 2 × 2 matrices over the integers Mquaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion a ...

, 0 is the additive identity.
*In the ring of function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

s from R to R, the function mapping
Mapping may refer to:
* Mapping (cartography), the process of making a map
* Mapping (mathematics), a synonym for a mathematical function and its generalizations
** Mapping (logic), a synonym for functional predicate
Types of mapping
* Animated ...

every number to 0 is the additive identity.
*In the additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structure ...

of vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

s in Rzero vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

is the additive identity.
Properties

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 then follows from the above that : = + 0 = 0' + = .The additive identity annihilates ring elements

In a system with a multiplication operation that distributes over addition, the additive identity is a multiplicativeabsorbing elementIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

, meaning that for any ''s'' in ''S'', . This follows because:
:$\backslash begin\; s\; \backslash cdot\; 0\; \&=\; s\; \backslash cdot\; (0\; +\; 0)\; =\; s\; \backslash cdot\; 0\; +\; s\; \backslash cdot\; 0\; \backslash \backslash \; \backslash Rightarrow\; s\; \backslash cdot\; 0\; \&=\; s\; \backslash cdot\; 0\; -\; s\; \backslash cdot\; 0\; \backslash \backslash \; \backslash Rightarrow\; s\; \backslash cdot\; 0\; \&=\; 0.\; \backslash end$
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, i.e. 0 = 1. Let ''r'' be any element of ''R''. Then : ''r'' = ''r'' × 1 = ''r'' × 0 = 0 proving that ''R'' is trivial, i.e. ''R'' = . Thecontrapositive
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

, that if ''R'' is non-trivial then 0 is not equal to 1, is therefore shown.
See also

*0 (number)
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

*Additive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be ...

*Identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

*Multiplicative identity
In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...

References

Bibliography

*David S. Dummit, Richard M. Foote, ''Abstract Algebra'', Wiley (3rd ed.): 2003, .External links

*{{PlanetMath , urlname=UniquenessOfAdditiveIdentityInARing2 , title=Uniqueness of additive identity in a ring , id=5676 Abstract algebra Elementary algebra Group theory Ring theory 0 (number)