PlanetMath: ideal inverting in Prüfer ring

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ideal inverting in Prüfer ring

(Theorem)

Theorem 1 Let $\mathfrak{a}_1$ , ..., $\mathfrak{a}_n$ be invertible fractional ideals of a Prüfer ring. Then also their sum and intersection are invertible, and the inverse ideals of these are obtained by the formulae resembling de Morgan's laws:

$\displaystyle (\mathfrak{a}_1+\cdots+\mathfrak{a}_n)^{-1} = \mathfrak{a}_1^{-1}\cap\cdots\cap\mathfrak{a}_n^{-1}$

$\displaystyle (\mathfrak{a}_1\cap\cdots\cap\mathfrak{a}_n)^{-1} = \mathfrak{a}_1^{-1}+\cdots+\mathfrak{a}_n^{-1}$

This duality is due to the fact, that the sum of any ideals is the smallest ideal containing these ideals and the intersection of the ideals is the largest ideal contained in each of these ideals. Cf. sum of ideals, quotient of ideals.

Bibliography

J. Pahikkala: ``Some formulae for multiplying and inverting ideals''.

Annales universitatis turkuensis 183. Turun yliopisto (University of Turku) 1982.

"ideal inverting in Prüfer ring" is owned by pahio.

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Cross-references: quotient of ideals, sum of ideals, contained, ideals, duality, de Morgan's laws, inverse ideals, intersection, sum, Prüfer ring, fractional ideals, invertible
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This is version 8 of ideal inverting in Prüfer ring, born on 2004-08-15, modified 2006-09-27.
Object id is 6103, canonical name is IdealInvertingInPruferRing.
Accessed 1645 times total.

Classification:

AMS MSC:

13C13 (Commutative rings and algebras :: Theory of modules and ideals :: Other special types)

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