Z/nZ ring

In Mathematical, and more particularly in Algebra, Z /n Z is a particular case of ring.

Any unit ring contains either an isomorphous subring with Z /n Z or with Z the ring of the whole .

This ring plays a particular part in Arithmétique, it is indeed the basic tool of arithmetic modular.

The article Congruence on the entireties covers the same subject with a more didactic and less exhaustive approach, the article modular Arithmétique treats history of this concept, tools used as well as its applications.

Construction of Z / nZ

Ideals of Z

See also: Ideal

The Euclidean Division in Z watch that this unit is a Euclidean Anneau, consequently Z is a principal Anneau. That means that for any ideal I of Z , there exists an entirety N such as I is equal to nZ . As the ideals nZ and - nZ is confused, it is always possible to choose positive N . Subsequently of the article N indicates a positive entirety.

Ring quotient

See also: Ring quotient

The construction of Z / nZ corresponds to the general construction of the rings quotients. Here the relation of equivalence corresponds to traditional the Congruence on the entireties. An element of Z / nZ is the class of the elements having all the same remainder by Euclidean division by N .

An element is identified by a member of its class, often the entirety ranging between 0 and N - 1. It is sometimes noted $\backslash \; scriptstyle\; \backslash \; dowry\; a$ or $\backslash \; scriptstyle\; \backslash \; bar\; a$, thus in Z /6 Z $\backslash \; scriptstyle\; \backslash \; bar\; 2$ indicates the class containing the elements 2,8,14 etc… When there does not exist ambiguity, one uses simply the letter has .

• the elements of Z / nZ are called classes modulo N or residue .

Properties

Elementary properties

See also: Ring (mathematics)

The theory of the rings directly makes it possible to show certain properties of the ring.

* the ring Z / nZ is unit .

It is a direct consequence owing to the fact that Z is.

* the ring Z / nZ is principal and of Bézout .

A ring is principal if and only if all its ideals are principal. If a ring is principal, its quotient by an ideal is also principal, but Z is a principal ring. In practice and as for Z , all the Sub-group S additives and all the subrings are also principal ideals. If m is a divider of N then it exists a single ideal of isomorphous Z / nZ with Z / mZ , this result is a direct consequence of the third proposal of the paragraph fundamental Théorème of the article cyclic Groupe.

A ring is known as of Bézout if and only so for any element has and B having like dividers the common only invertible elements, it exists two elements α and β such as α. has + β. B = 1. Z / nZ is a ring of Bézout because any principal ring is.

If N is not first, then the ring Z / nZ is not just, it is thus neither Euclidean nor factorial.

Additive structure

See also: cyclic Group

The structure of the group ( Z / nZ ) is that of a group monogene, i.e. generated by a single element. If N is equal to 0 one obtains an isomorphous group with Z and with any monogene group of an infinite nature.

If N is different from 0, then the group is cyclic, its structure is clarified in the detailed article.

Chinese theorem

See also: Theorem of the Chinese remainders

The logic of the Chinese theorem still applies, thus the properties of the paragraph Chinese Théorème of the cyclic article Groupe still apply. It is enough to check them to validate that the Morphisme of group used is also a morphism of ring.

* Is U and v two whole first between them, then the ring Z / U . vZ is isomorphous with the produced of the rings of order Z / uZ and Z / vZ .

Note : If U and v is not first between them, then the produced ring does not contain an element of a nature higher than the ppcm of U and v . This ring is thus not isomorphous with the ring Z / U . vZ .

This proposal involves a single decomposition of Z / nZ in factors first. The fundamental Théorème of arithmetic the watch that N breaks up in the following single way:

$n\; =\; \backslash \; prod\_\; \{i=1\}\; ^k\; p\_i^\; \{\backslash \; alpha\_i\}\; \backslash ;$ Or ( p _i) is a family of K prime numbers all distinct and α_i from the entireties equal to or higher than one. The powers of the prime numbers of the product are very first between them. A simple recurrence shows:

* Z / nZ breaks up in a single way into a product of rings quotients of Z of cardinal a power of a prime number.

Case where Z / nZ is a body

See also: Body (mathematical)

* Z /n Z is a body if and only if N is a Prime number .

Indeed, this proposal is a direct consequence of the Identité of Bézout. Let us suppose N first, then if has is an entirety first with N , i.e. not multiple of N , there exist two entireties B and C such as: $ab\; +\; nc\; =\; 1\; \backslash ;$ What means that the class of has is invertible of reverse the class of B .

Reciprocally if N is not first, there exist two entireties has and B different of N and 1 such as their product is equal to N . The class of has as well as the class of B are dividers of zero, which does not exist in a body.

Characteristic of a ring

See also: Characteristic of a ring

Either has a unit ring, it exists a single morphism of ring φ of Z in has which with 1_Z associates 1_A. Either N positive entirety such as the core of φ or equal to nZ . The canonical decomposition of φ (cf the paragraph Morphisme of ring of the article Idéal) watch which there exists a subring of has isomorphous with Z / nZ .

• the entirety N is called characteristic of the ring has .

Thus, any unit ring contains an isomorphous subring is with Z if N is equal to 0, that is to say with Z / nZ . It is one of the reasons which makes this family of ring interesting.

Group units

See also: Group of the units

The group of the units of a ring corresponds to the formed multiplicative group of the invertible elements. Such elements are called unit .

* Is m an entirety, its class is a unit if and only if m is first with N .

If m is first with N then it is invertible, if not either D a common divider different from one, or K entirety such as d.k = N , the fact that m.k or a multiple of N watch that m is a divider of zero and thus is noninvertible.

* the order of the group of the units is equal to φ ( N ) if φ indicates the function Indicatrice of Euler.

An element of the additive group Z / nZ is generating if and only if it is first with N , because its order is then equal to N . However the paragraph Indicating of Euler of the cyclic article Group watch which the number of generating elements is equal to φ ( N ).

Case where N is first

See also: Exhibitor of a group

If N is first i.e. if the ring is a body, the structure is the following one:

* If N is a prime number, the group of the units of the body Z / nZ is a cyclic group of order N - 1.

Indeed, any element other than that no one is invertible, the order of the multiplicative group is thus N - 1. The multiplicative group is naturally finished, it admits an exhibitor E , the exhibitor is the lowest common multiple of the orders of the various elements of the multiplicative group. Let us consider the Polynôme Z / nZ according to: X^e - 1. He thus admits for roots all the elements of the multiplicative group N - 1 different roots. However any polynomial with coefficients in a body has a degree equal to or higher than its number of roots. One from of deduced that E is equal to or higher than N - 1. Theorem of Lagrange, which has as a corollary the fact that the order of an element is a divider about the group, shows that E is equal to N - 1.

To conclude it is enough to note that any finished abelian group has an element of order the exhibitor, this property is shown in the detailed article. The multiplicative group has an element of order the cardinal of the group and which is thus primitive, which shows that the group is cyclic and finishes the demonstration.

Case where N is not first

If N is not first, the structure is naturally that of a abelian Groupe finished it thus corresponds to a product of cyclic groups according to the Théorème of Kronecker. The structure is more complex than that of the preceding case, several proposals are necessary to clarify it.

* Is N and m two entireties first between them, the group of the units of Z / n.mZ is isomorphous with the direct produced of the groups of the units of Z / nZ and of Z / mZ .

It is a consequence of the Chinese theorem.

The fundamental theorem of the arithmetic limit then the study with the case or N is equal to p ^r with p a prime number and R a strictly positive entirety. Two configurations arise:

* If p is equal to two and R equal to or higher than three, the group of the units is the direct product of a group of a nature two generated by the class of -1 and of a cyclic group generated by the class of 5.

* If p is different from two, then the group of the units is cyclic.

All the cases are not treated, there remains that or p is equal to two and R is equal to one or two. However these cases are commonplace, the group contains one or two elements and consequently is cyclic.

See too

External bonds

• Whole modular by modular David A. Madore
• Arithmetic by B. Perrin Riou
• Z '' N '' Z on the mathématiques.net

References

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