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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
image ideal of divisor
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(Theorem)
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Theorem. If an integral domain
 has a divisor theory
 , then the subset
![$ [\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/10568/l2h/img3.png "$ [\mathfrak{a}]$") of
 , consisting of 0 and all elements divisible by a divisor
 , is an ideal of
 . The mapping
from the set
 of divisors into the set of ideals of
 is injective and maps any principal divisor  to the principal ideal  .
Proof. Let
![$ \alpha,\,\beta \in [\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/10568/l2h/img12.png "$ \alpha,\,\beta \in [\mathfrak{a}]$") and
 . Then, by the postulate 2 of divisor theory,
 is divisible by
 or is 0, and in both cases belongs to
![$ [\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/10568/l2h/img16.png "$ [\mathfrak{a}]$") . When
 , we can write
 with
 a divisor. According to the homomorphicity of the mapping
 , we have
and therefore the element
 is divisible by
 , i.e.
![$ \lambda\alpha \in [\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/10568/l2h/img24.png "$ \lambda\alpha \in [\mathfrak{a}]$") . Thus,
![$ [\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/10568/l2h/img25.png "$ [\mathfrak{a}]$") is an ideal of
 .
The injectivity of the mapping
![$ \mathfrak{a} \mapsto [\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/10568/l2h/img27.png "$ \mathfrak{a} \mapsto [\mathfrak{a}]$") follows from the postulate 3 of divisor theory.
The ideal
![$ [\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/10568/l2h/img28.png "$ [\mathfrak{a}]$") may be called the image ideal of
 or the ideal determined by the divisor
 .
Remark. There are integral domains
 having a divisor theory but also having ideals which are not of the form
![$ [\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/10568/l2h/img32.png "$ [\mathfrak{a}]$") (for example a polynomial ring in two indeterminates and its ideal formed by the polynomials without constant term). Such rings have “too many ideals”. On the other hand, in some integral domains the monoid of principal ideals cannot be embedded into a free monoid; thus those rings cannot have a divisor theory.
- 1
- М. М. Постников: Введение в теорию алгебраических чисел. Издательство ``Наука''. Москва(1982).
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"image ideal of divisor" is owned by pahio.
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(view preamble)
| Also defines: |
image ideal, ideal determined by the divisor |
This object's parent.
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Cross-references: free monoid, monoid, rings, constant term, polynomials, indeterminates, polynomial ring, postulate, proof, principal ideal, principal divisor, maps, injective, mapping, ideal, divisor, divisible, subset, divisor theory, integral domain
There are 2 references to this entry.
This is version 6 of image ideal of divisor, born on 2008-05-06, modified 2008-05-07.
Object id is 10568, canonical name is ImageIdealOfDivisor.
Accessed 482 times total.
Classification:
| AMS MSC: | 11A51 (Number theory :: Elementary number theory :: Factorization; primality) | | | 13A05 (Commutative rings and algebras :: General commutative ring theory :: Divisibility) | | | 13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory) |
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Pending Errata and Addenda
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![$\displaystyle \mathfrak{a} \mapsto [\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/10568/l2h/img7.png "$\displaystyle \mathfrak{a} \mapsto [\mathfrak{a}]$"){kind=small}
