ℤ_n

ℤ

n

Let

n

∈

ℤ

. An

equivalence relation

, called

congruence

(

http://planetmath.org/Congruent2

), can be defined on

ℤ

by

a

≡

b

⁢

mod

⁡

n

iff

n

divides

b

-

a

. Note first of all that

a

≡

b

⁢

mod

⁡

n

iff

a

≡

b

⁢

mod

⁡

|

n

|

. Thus, without loss of generality, only nonnegative

n

need be considered. Secondly, note that the case

n

=

0

is not very interesting. If

a

≡

b

⁢

mod

⁡

0

, then

0

divides

b

-

a

, which occurs exactly when

a

=

b

. In this case, the set of all

equivalence classes

can be identified with

ℤ

. Thus, only

positive

n

need be considered. The set of all equivalence classes of

ℤ

under the given equivalence relation is called

ℤ

n

.

Some mathematicians consider the notation

ℤ

n

to be archaic and somewhat confusing. This matter of notation is most considerable when

n

=

p

for some prime (

http://planetmath.org/PrimeNumber

)

p

, as

ℤ

p

is used to refer to the

p

-adic

integers

(

http://planetmath.org/MathbbZ_p

). To avoid this confusion, some mathematicians use the notation

ℤ

/

n

⁢

ℤ

instead of

ℤ

n

. On the other hand, the notation

ℤ

n

should not cause confusion when

n

is not prime, and is an intuitive shorthand way to write

ℤ

/

n

⁢

ℤ

. Thus, others use

𝔽

p

when

n

=

p

for some prime

p

and

ℤ

n

otherwise. (The explanation of the usage of

𝔽

p

will come later.) Still others, especially those who are unfamiliar with the , use the notation

ℤ

n

exclusively. (In this entry, the notation

ℤ

n

is used exclusively, though it is highly recommended to use another notation when

n

=

p

for some prime

p

.)

One usually identifies an element of

ℤ

n

(which is technically a class (

http://planetmath.org/EquivalenceClass

), ) with the unique

element

r

in the class such that

0

≤

r

<

n

. One can use the

division algorithm

to establish that, for each class, an

r

as described exists uniquely. (The set of all

r

’s as described is an example of a

residue system

.) Thus, the sets

ℤ

n

are finite with exactly

n

elements.

Addition

and

multiplication

operations

can also be defined on

ℤ

n

in a natural way that corresponds to the operations on

ℤ

. Under these operations,

ℤ

n

is a

commutative ring

with

identity

(

http://planetmath.org/MultiplicativeIdentity

) as well as a

cyclic ring

with

behavior

1

. When

n

=

p

for some prime

p

,

ℤ

n

is a field. In this case, the notation

𝔽

p

highlights the fact that the is a field. When

n

is

composite

,

ℤ

n

has

zero divisors

and thus is neither a field nor an

integral domain

. Also note that

ℤ

1

is a

zero ring

, since all integers are

equivalent

(

http://planetmath.org/Equivalent

), yielding only one equivalence class.

The

n

in both

ℤ

n

and

a

≡

b

⁢

mod

⁡

n

is called the

modulus

. Performing computations such as addition,

subtraction

, multiplication, and taking exponents (

http://planetmath.org/Exponent2

) in one of the rings

ℤ

n

is called

modular arithmetic

.

Title

ℤ

n

Canonical

name

mathbbZn

Date of creation

2013-03-22 15:58:10

Last modified on

2013-03-22 15:58:10

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

25

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 13-00

Classification

msc 13M05

Classification

msc 11-00

Synonym

integers mod n

Related topic

ResidueSystems

Related topic

MathbbZ

Related topic

CyclicRingsThatAreIsomorphicToKmathbbZ_kn

Related topic

Congruences

Related topic

EquivalenceRelation

Defines

modulus

Defines

modular arithmetic

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