Editing Integer (section)

== Algebraic properties ==
[[File:Number-line.svg|right|thumb|300px|Integers can be thought of as discrete, equally spaced points on an infinitely long [[number line]]. In the above, non-[[Sign (mathematics)#Terminology for signs|negative]] integers are shown in blue and negative integers in red.]]
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Like the [[natural numbers]], <math>\mathbb{Z}</math> is [[closure (mathematics)|closed]] under the [[binary operation|operations]] of addition and [[multiplication]], that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, {{num|0}}), <math>\mathbb{Z}</math>, unlike the natural numbers, is also closed under [[subtraction]].<ref>{{Cite web|title=Integer {{!}} mathematics|url=https://www.britannica.com/science/integer|access-date=2020-08-11|website=Encyclopedia Britannica|language=en}}</ref>

The integers form a [[ring (mathematics)|ring]] which is the most basic one, in the following sense: for any ring, there is a unique [[ring homomorphism]] from the integers into this ring. This [[universal property]], namely to be an [[initial object]] in the [[category of rings]], characterizes the ring&nbsp;<math>\mathbb{Z}</math>. This unique homomorphism is [[injective]] if and only if the [[characteristic (algebra)|characteristic]] of the ring is zero. It follows that every ring of characteristic zero contains a subring isomorphic to {{tmath|\Z}}, which is its smallest subring.

<math>\mathbb{Z}</math> is not closed under [[division (mathematics)|division]], since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under [[exponentiation]], the integers are not (since the result can be a fraction when the exponent is negative).

The following table lists some of the basic properties of addition and multiplication for any integers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}:
{|class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+Properties of addition and multiplication on integers
!
!scope="col" |Addition
!scope="col" |Multiplication
|-
!scope="row" |[[Closure (mathematics)|Closure]]:
|{{math|''a'' + ''b''}} is an integer
|{{math|''a'' × ''b''}} is an integer
|-
!scope="row"|[[Associativity]]:
|{{math|''a'' + (''b'' + ''c'') {{=}} (''a'' + ''b'') + ''c''}}
|{{math|''a'' × (''b'' × ''c'') {{=}} (''a'' × ''b'') × ''c''}}
|-
!scope="row" |[[Commutativity]]:
|{{math|''a'' + ''b'' {{=}} ''b'' + ''a''}}
|{{math|''a'' × ''b'' {{=}} ''b'' × ''a''}}
|-
!scope="row" |Existence of an [[identity element]]:
|{{math|''a'' + 0 {{=}} ''a''}}
|{{math|''a'' × 1 {{=}} ''a''}}
|-
!scope="row" |Existence of [[inverse element]]s:
|{{math|''a'' + (−''a'') {{=}} 0}}
|The only invertible integers (called [[Unit (ring theory)|units]]) are −1 and 1.
|-
!scope="row" |[[Distributivity]]:
|colspan=2 align=center |{{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}} and {{math|(''a'' + ''b'') × ''c'' {{=}} (''a'' × ''c'') + (''b'' × ''c'')}}
|-
!scope="row" |No [[zero divisor]]s:
| || | If {{math|''a'' × ''b'' {{=}} 0}}, then {{math|''a'' {{=}} 0}} or {{math|''b'' {{=}} 0}} (or both)
|}

The first five properties listed above for addition say that <math>\mathbb{Z}</math>, under addition, is an [[abelian group]]. It is also a [[cyclic group]], since every non-zero integer can be written as a finite sum {{nowrap|1 + 1 + ... + 1}} or {{nowrap|(−1) + (−1) + ... + (−1)}}. In fact, <math>\mathbb{Z}</math> under addition is the ''only'' infinite cyclic group—in the sense that any infinite cyclic group is [[group isomorphism|isomorphic]] to <math>\mathbb{Z}</math>.

The first four properties listed above for multiplication say that <math>\mathbb{Z}</math> under multiplication is a [[commutative monoid]]. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that <math>\mathbb{Z}</math> under multiplication is not a group.

All the rules from the above property table (except for the last), when taken together, say that <math>\mathbb{Z}</math> together with addition and multiplication is a [[commutative ring]] with [[multiplicative identity|unity]]. It is the prototype of all objects of such [[algebraic structure]]. Only those [[equality (mathematics)|equalities]] of [[algebraic expression|expressions]] are true in&nbsp;<math>\mathbb{Z}</math> [[for all]] values of variables, which are true in any unital commutative ring. Certain non-zero integers map to [[additive identity|zero]] in certain rings.

The lack of [[zero divisor]]s in the integers (last property in the table) means that the commutative ring&nbsp;<math>\mathbb{Z}</math> is an [[integral domain]].

The lack of multiplicative inverses, which is equivalent to the fact that <math>\mathbb{Z}</math> is not closed under division, means that <math>\mathbb{Z}</math> is ''not'' a [[field (mathematics)|field]]. The smallest field containing the integers as a [[subring]] is the field of [[rational number]]s. The process of constructing the rationals from the integers can be mimicked to form the [[field of fractions]] of any integral domain. And back, starting from an [[algebraic number field]] (an extension of rational numbers), its [[ring of integers]] can be extracted, which includes <math>\mathbb{Z}</math> as its [[subring]].

Although ordinary division is not defined on <math>\mathbb{Z}</math>, the division "with remainder" is defined on them. It is called [[Euclidean division]], and possesses the following important property: given two integers {{math|''a''}} and {{math|''b''}} with {{math|''b'' ≠ 0}}, there exist unique integers {{math|''q''}} and {{math|''r''}} such that {{math|''a'' {{=}} ''q'' × ''b'' + ''r''}} and {{math|0 ≤ ''r'' <  {{!}}''b''{{!}}}}, where {{math|{{!}}''b''{{!}}}} denotes the [[absolute value]] of {{math|''b''}}. The integer {{math|''q''}} is called the ''quotient'' and {{math|''r''}} is called the ''[[remainder]]'' of the division of {{math|''a''}} by {{math|''b''}}. The [[Euclidean algorithm]] for computing [[greatest common divisor]]s works by a sequence of Euclidean divisions.

The above says that <math>\mathbb{Z}</math> is a [[Euclidean domain]]. This implies that <math>\mathbb{Z}</math> is a [[principal ideal domain]], and any positive integer can be written as the products of [[prime number|primes]] in an [[essentially unique]] way.<ref>{{cite book |first=Serge |last=Lang |author-link=Serge Lang |title=Algebra |edition=3rd |publisher=Addison-Wesley |year=1993 |isbn=978-0-201-55540-0 |pages=86–87}}</ref> This is the [[fundamental theorem of arithmetic]].