Editing Modular arithmetic (section)

== Integers modulo ''m'' ==
In the context of this paragraph, the modulus {{math|''m''}} is almost always taken as positive.

The set of all [[#Congruence classes|congruence classes]] modulo {{math|''m''}} is a [[ring (mathematics)|ring]] called the '''ring of integers modulo {{math|''m''}}''', and is denoted <math display=inline>\mathbb{Z}/m\mathbb{Z}</math>, <math>\mathbb{Z}/m</math>, or <math>\mathbb{Z}_m</math>.<ref>{{Cite web|date=2013-11-16|title=2.3: Integers Modulo n|url=https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Book%3A_Introduction_to_Algebraic_Structures_(Denton)/02%3A_Groups_I/2.03%3A_Integers_Modulo_n|access-date=2020-08-12|website=Mathematics LibreTexts|language=en|archive-date=2021-04-19|archive-url=https://web.archive.org/web/20210419035455/https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Book%3A_Introduction_to_Algebraic_Structures_(Denton)/02%3A_Groups_I/2.03%3A_Integers_Modulo_n|url-status=live}}</ref> 
The ring <math>\mathbb{Z}/m\mathbb{Z}</math> is fundamental to various branches of mathematics (see ''{{section link|#Applications}}'' below).
(In some parts of [[number theory]] the notation <math>\mathbb{Z}_m</math> is avoided because it can be confused with the set of [[P-adic integer|{{math|''m''}}-adic integers]].) 

For {{math|''m'' > 0}} one has
: <math>\mathbb{Z}/m\mathbb{Z} = \left\{ \overline{a}_m \mid a \in \mathbb{Z}\right\} = \left\{ \overline{0}_m, \overline{1}_m, \overline{2}_m,\ldots, \overline{m{-}1}_m \right\}.</math>

When {{math|1=''m'' = 1}}, <math>\mathbb{Z}/m\mathbb{Z}</math> is the [[zero ring]]; when {{math|1=''m'' = 0}}, <math>\mathbb{Z}/m\mathbb{Z}</math> is not an [[empty set]]; rather, it is [[isomorphism|isomorphic]] to <math>\mathbb{Z}</math>, since {{math|1={{overline|''a''}}<sub>0</sub> = {{mset|''a''}}}}.

Addition, subtraction, and multiplication are defined on <math>\mathbb{Z}/m\mathbb{Z}</math> by the following rules:
* <math>\overline{a}_m + \overline{b}_m = \overline{(a + b)}_m</math>
* <math>\overline{a}_m - \overline{b}_m = \overline{(a - b)}_m</math>
* <math>\overline{a}_m \overline{b}_m = \overline{(a b)}_m.</math>

The properties given before imply that, with these operations, <math>\mathbb{Z}/m\mathbb{Z}</math> is a [[commutative ring]]. For example, in the ring <math>\mathbb{Z}/24\mathbb{Z}</math>, one has
: <math>\overline{12}_{24} + \overline{21}_{24} = \overline{33}_{24}= \overline{9}_{24}</math>
as in the arithmetic for the 24-hour clock.

The notation <math>\mathbb{Z}/m\mathbb{Z}</math> is used because this ring is the [[quotient ring]] of <math>\mathbb{Z}</math> by the [[ideal (ring theory)|ideal]] <math>m\mathbb{Z}</math>, the set formed by all multiples of {{math|''m''}}, i.e., all numbers {{math|''k m''}} with <math>k\in\mathbb{Z}.</math>

Under addition, <math>\mathbb Z/m\Z</math> is a [[cyclic group]].  All finite cyclic groups are isomorphic with <math>\mathbb Z/m\mathbb Z</math> for some {{mvar|m}}.<ref>Sengadir T., {{Google books|id=nglisrt9IewC|page=293|text=Zn is generated by 1|title=Discrete Mathematics and Combinatorics}}</ref>

The ring of integers modulo {{math|''m''}} is a [[field (mathematics)|field]], i.e., every nonzero element has a [[Modular multiplicative inverse|multiplicative inverse]], if and only if {{math|''m''}} is [[Prime number|prime]]. If {{math|1=''m'' = ''p''{{i sup|''k''}}}} is a [[prime power]] with {{math|''k'' > 1}}, there exists a unique (up to isomorphism) finite field <math>\mathrm{GF}(m) =\mathbb F_m</math> with {{math|''m''}} elements, which is ''not'' isomorphic to <math>\mathbb Z/m\mathbb Z</math>, which fails to be a field because it has [[zero-divisor]]s.

If {{math|''m'' > 1}}, <math>(\mathbb Z/m\mathbb Z)^\times</math> denotes the [[multiplicative group of integers modulo n|multiplicative group of the integers modulo {{math|''m''}}]] that are invertible. It consists of the congruence classes {{math|{{overline|''a''}}{{sub|''m''}}}}, where {{math|''a''}} [[coprime integers|is coprime]] to {{math|''m''}}; these are precisely the classes possessing a multiplicative inverse. They form an [[abelian group]] under multiplication; its order is {{math|''φ''(''m'')}}, where {{mvar|φ}} is [[Euler's totient function]].