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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
invertible ideal is finitely generated
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(Theorem)
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Proof. Let  be the total ring of fractions of  and  the unity of  .We first show that the inverse ideal of
 has the unique quotient presentation
![$ [R':\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/7239/l2h/img9.png "$ [R':\mathfrak{a}]$") where
 . If
 is an inverse ideal of
 , it means that
 . Therefore we have
so that
This implies that
![$ \mathfrak{aa}^{-1} = \mathfrak{a}[R'\!:\!\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/7239/l2h/img16.png "$ \mathfrak{aa}^{-1} = \mathfrak{a}[R'\!:\!\mathfrak{a}]$") , and because
 is a cancellation ideal, it must mean that
![$ \mathfrak{a}^{-1} = [R'\!:\!\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/7239/l2h/img18.png "$ \mathfrak{a}^{-1} = [R'\!:\!\mathfrak{a}]$") , i.e.
![$ [R'\!:\!\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/7239/l2h/img19.png "$ [R'\!:\!\mathfrak{a}]$") is the unique inverse of the ideal
 .
Since
![$ \mathfrak{a}[R'\!:\!\mathfrak{a}] = R'$](http://images.planetmath.org:8080/cache/objects/7239/l2h/img21.png "$ \mathfrak{a}[R'\!:\!\mathfrak{a}] = R'$") , there exist some elements
 of
 and the elements
 of
![$ [R'\!:\!\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/7239/l2h/img25.png "$ [R'\!:\!\mathfrak{a}]$") such that
 . Then an arbitrary element  of
 satisfies
because every  belongs to the ring  . Accordingly,
 . Since the converse inclusion is apparent, we have seen that
 is a finite generator system of the invertible ideal
 .
Since the elements  belong to the total ring of fractions of  , we can choose such a regular element  of  that each of the products  belongs to  . Then
and thus the fractional ideal
 contains a regular element of  , which obviously is regular in  , too.
- 1
- R. GILMER: Multiplicative ideal theory. Queens University Press. Kingston, Ontario (1968).
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"invertible ideal is finitely generated" is owned by pahio.
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(view preamble)
Cross-references: regular, contains, products, invertible ideal, finite, inclusion, converse, ring, ideal, inverse, cancellation ideal, implies, inverse ideal, unity, total ring of fractions, proof, finitely generated, fractional ideal, regular elements, commutative ring
There are 2 references to this entry.
This is version 6 of invertible ideal is finitely generated, born on 2005-07-19, modified 2006-10-03.
Object id is 7239, canonical name is InvertibleIdealIsFinitelyGenerated.
Accessed 1463 times total.
Classification:
| AMS MSC: | 13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization) |
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Pending Errata and Addenda
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![$\displaystyle \mathfrak{a}^{-1} \subseteq \{t\in T\,\vdots \,\,\, t\mathfrak{a}\subseteq R'\} = [R'\!:\!\mathfrak{a}],$](http://images.planetmath.org:8080/cache/objects/7239/l2h/img14.png "$\displaystyle \mathfrak{a}^{-1} \subseteq \{t\in T\,\vdots \,\,\, t\mathfrak{a}\subseteq R'\} = [R'\!:\!\mathfrak{a}],$")
![$\displaystyle R' = \mathfrak{aa}^{-1} \subseteq \mathfrak{a}[R'\!:\!\mathfrak{a}] \subseteq R'.$](http://images.planetmath.org:8080/cache/objects/7239/l2h/img15.png "$\displaystyle R' = \mathfrak{aa}^{-1} \subseteq \mathfrak{a}[R'\!:\!\mathfrak{a}] \subseteq R'.$"){kind=small}
