implications of having divisor theory

implications of having divisor theory

The existence of a

divisor theory

restricts strongly the of an

integral domain

, as is seen from the following propositions.

Proposition 1.

An integral domain

𝒪

which has a divisor theory

𝒪

*

→

𝔇

,  is

integrally closed

in its

quotient field

.

Proof.

Let

ξ

be an element of the quotient field of

𝒪

which is integral over

𝒪

.  Then

ξ

satisfies an

equation

ξ

n

+

α

1

⁢

ξ

n

-

1

+

…

+

α

n

=

0

(1)

where

α

1

,

…

,

α

n

∈

𝒪

.  Now, we can write

ξ

=

ϰ

λ

with

ϰ

,

λ

∈

𝒪

,  whence (1) may be written

ϰ

n

=

-

α

1

⁢

λ

⁢

ϰ

n

-

1

-

α

2

⁢

λ

2

⁢

ϰ

n

-

2

-

…

-

α

n

⁢

λ

n

.

(2)

Let us make the antithesis that

ξ

does not belong to

𝒪

itself.  Then

λ

∤

ϰ

and therefore we have for the corresponding principal divisors

(

λ

)

∤

(

ϰ

)

.  We infer that there is a

prime divisor

factor

𝔭

of

(

λ

)

and an

integer

k

≧

0

such that

𝔭

k

∣

(

ϰ

)

,

𝔭

k

+

1

∤

(

ϰ

)

,

𝔭

k

+

1

∣

(

λ

)

.

By the condition 2 of the definition of divisor theory (

http://planetmath.org/DivisorTheory

), the right hand side of the equation (2) is

divisible

by

𝔭

(

k

+

1

)

+

(

n

-

1

)

⁢

k

=

𝔭

k

⁢

n

+

1

.

On the other side, the highest power of

𝔭

, by which the

divisor

(

ϰ

n

)

is divisible, is

𝔭

k

⁢

n

.  Accordingly, the different sides of (2) show different divisibility by powers of

𝔭

.  This contradictory situation means that the antithesis was wrong and thus the proposition has been proven.

Proposition 2.

When an integral domain

𝒪

has a divisor theory

𝒪

*

→

𝔇

,  then each element of

𝒪

*

has only a finite number of non-associated (

http://planetmath.org/

Associates

) factors (

http://planetmath.org/DivisibilityInRings

).

Proof.

Let

ξ

be an arbitrary non-zero element of

𝒪

.  We form the

prime factor

presentation

of the corresponding principal divisor

(

ξ

)

:

(

ξ

)

=

𝔭

1

⁢

𝔭

2

⁢

⋯

⁢

𝔭

r

This is unique up to the

ordering

of the factors;

r

≧

0

.  Then we form of the prime divisors

𝔭

i

all products having

k

factors
(

0

≦

k

≦

r

) and choose from the products those which are principal divisors.  Thus we obtain a set of factors of

(

ξ

)

containing at most

(

r

k

)

elements.  All different principal divisor factors of

(

ξ

)

are gotten, as

k

runs all integers from 0 to

r

, and their number is at most equal to

∑

k

=

0

r

(

r

k

)

=

2

r

(see 5. in the

binomial coefficients

).  To every principal divisor , there corresponds a class (

http://planetmath.org/EquivalenceClass

) of associate factors of

ξ

, and the elements of distinct classes (

http://planetmath.org/EquivalenceClass

) are non-associates.  Since

ξ

has not other factors, the number of its non-associated factors is at most

2

r

.

Title

implications of having divisor theory

Canonical name

ImplicationsOfHavingDivisorTheory

Date of creation

2013-03-22 17:59:13

Last modified on

2013-03-22 17:59:13

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Topic

Classification

msc 11A51

Classification

msc 13A05

Synonym

properties of rings having a divisor theory

Related topic

DivisorTheoryAndExponentValuations

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