divisor theory

divisor theory

0.1

Divisibility in a monoid

In a

commutative monoid

𝔇

, one can speak of

divisibility

: its

element

𝔞

is

divisible

by its element

𝔟

, iff

𝔞

=

𝔟

⁢

𝔠

where

𝔠

∈

𝔇

.  An element

𝔭

of

𝔇

, distinct from the unity

𝔢

of

𝔇

, is called a

prime element

of

𝔇

, when

𝔭

is divisible only by itself and

𝔢

.  The monoid

𝔇

has a

unique prime

factorisation

, if every element

𝔞

of

𝔇

can be presented as a finite

product

𝔞

=

𝔭

1

⁢

𝔭

2

⁢

⋯

⁢

𝔭

r

of prime elements and this is unique up to the

𝔭

i

; then we may say that

𝔇

is a

free monoid

on the set of its prime elements.

If the monoid

𝔇

has a unique prime factorisation,

𝔢

is divisible only by itself.  Two elements of

𝔇

have always a

greatest common factor

.  If a product

𝔞

⁢

𝔟

is divisible by a prime element

𝔭

, then at least one of

𝔞

and

𝔟

is divisible by

𝔭

.

0.2

Divisor theory of an integral domain

Let

𝒪

be an

integral domain

and

𝒪

*

the set of its non-zero elements; this set forms a commutative monoid (with

identity

1) with respect to the

multiplication

of

𝒪

.  We say that the integral domain

𝒪

has a

divisor theory

, if there is a commutative monoid

𝔇

with unique prime factorisation and a

homomorphism

α

↦

(

α

)

from the monoid

𝒪

*

into the monoid

𝔇

, such that the following three properties are true:

1.

A divisibility (

http://planetmath.org/DivisibilityInRings

)

α

∣

β

in

𝒪

*

is valid iff the divisibility

(

α

)

∣

(

β

)

is valid in

𝔇

.

2.

If the elements

α

and

β

of

𝒪

*

are divisible by an element

𝔠

of

𝔇

, then also

α

±

β

are divisible by

𝔠

(‘‘

𝔠

∣

α

’’  means that

𝔠

∣

(

α

)

;  in , 0 is divisible by every element of

𝔇

).

3.

If

{

α

∈

𝒪

⁢

⋮

⁢

𝔞

∣

α

}

=

{

β

∈

𝒪

⁢

⋮

⁢

𝔟

∣

β

}

,  then

𝔞

=

𝔟

.

A divisor theory of

𝒪

is denoted by

𝒪

*

→

𝔇

.  The elements of

𝔇

are called

divisors

and especially the divisors of the form

(

α

)

, where

α

∈

𝒪

*

,

principal divisors

.  The prime elements of

𝔇

are

prime divisors

.

By 1, it is easily seen that two principal divisors

(

α

)

and

(

β

)

are equal iff the elements

α

and

β

are

associates

of each other.  Especially, the units of

𝒪

determine the

unit divisor

𝔢

.

0.3

Uniqueness theorems

Theorem

1.

An integral domain

𝒪

has at most one divisor theory.  In other words, for any pair of divisor theories

𝒪

*

→

𝔇

and

𝒪

*

→

𝔇

′

, there is an

isomorphism

φ

:

𝔇

→

𝔇

′

such that

φ

⁢

(

(

α

)

)

=

(

α

)

′

always when the principal divisors

(

α

)

∈

𝔇

and

(

α

)

′

∈

𝔇

′

correspond to the same element

α

of

𝒪

*

.

Theorem 2.

An integral domain

𝒪

is a unique factorisation domain (

http://planetmath.org/UFD

) if and only if

𝒪

has a divisor theory

𝒪

*

→

𝔇

in which all divisors are principal divisors.

Theorem 3.

If the divisor theory

𝒪

*

→

𝔇

comprises only a finite number of prime divisors, then

𝒪

is a unique factorisation domain.

The proofs of those theorems are found in [1], which is available also in Russian (original), English and French.

References

1

S. Borewicz & I. Safarevic

:

Zahlentheorie

.  Birkhäuser Verlag. Basel und Stuttgart (1966).

2

М. М. Постников:

Введение  в  теорию  алгебраических
чисел

.  Издательство
‘‘Наука’’. Москва (1982).

Title

divisor theory

Canonical name

DivisorTheory

Date of creation

2013-03-22 17:59:03

Last modified on

2013-03-22 17:59:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Definition

Classification

msc 11A51

Classification

msc 13A05

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Defines

divisor

Defines

prime divisor

Defines

principal divisor

Defines

unit divisor

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